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MATHEMATICS LIBRARY 
THE UNIVERSITY 


OF ILLINOIS 
LIBRARY 


The 
Frank Hall collection 
of arithmetics, 
presented by Professor 
H. L. Rietz of the 
University of Iowa. 
AOA SIF 


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Return this book on or before the 
Latest Date stamped below. 


University of Illinois Library 


L161—H41 


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APPLETONS’ STANDARD ARITHMETICS 


NUMBERS ILLUSTRATED 


AND APPLIED IN 


LANGUAGE, DRAWING, AND READING 
LESSONS 


AN ARITHMETIC FOR PRIMARY SCHOOLS 


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NEW YORK, BOSTON, AND CHICAGO 
D. APPLETON AND COMPANY 
1886 


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PREFACE. 


Ir is the design of this book, in the first place, to familiarize the chiid 
with numbers and their combinations, not by means of repeating such for- 
mule as 4 and 3 are 7, but by provoking observation to lead him to the adop- 
tion of the formula as a statement of his own experience. In this way an 
intimate and spontaneous association of thought and expression will de in- 
duced, and that listlessness avoided which makes it possible for many chil- 
dren to repeat 4 and 3 are 7, without a thought of 4 or 3, or of the com- 
bination of 4 and 3. 

It is not difficult to understand how it is that so much effort is wasted 
in teaching a child to answer always with readiness and confidence the 
question, “‘ How many are 4 and 3?” when it is recollected that the custom 
has too often been, not to lead him to tell us what he has observed and 
knows, but to repeat a form of words which he had committed to memory 
as mere words. The methods of onr schools are happily greatly improved 
in this respect, and it is the purpose of this book, especially of Part I, to 
afford a great variety of exercises in which the pupil may gain a wide ex- 
perience in the application of number to objects, and a ready perception of 
their relation to each other. 

The pictorial illustrations at the head of the pages, entitled ‘* The 
Conversation,” are designed for language-lessons in which the immediate - 
design is to excite thought and cultivate expression, but their adaptation to 
the ultimate purpose of the book will be readily understood. 

In the smaller illustrations, under the heading “‘ What can you tell?” the 
imagination is called into more active play, and the child is led to give more 
independent and original expression to the ideas gained from the pictures 
than the purpose of the foregoing exercises would permit, which was to 
follow out a logical and consistent plan of development. Here he is to find 
for himself the thread of the “story” hinted at in the picture, and weave 
it into a connected form for himself, the basis being the special combina- 
tions suggested in the picture. 

In these illustrations, the operations in numbers are more definitely 
brought to notice than previously, but it is especially desirable that po form 
of words or even process of objective illustration be repeated with such a 


4.63940 


4. 7E PREFACE. 


degree of uniformity that the mere form may assume undue importance, or 
supersede the exercise of intelligence. 

The Slates are to supplement and carry on the object-work, suggestions 
for which will be found in the general notes. They serve the twofold pur- 
pose: first, of teaching the child the use of the slate, at the same time 
familiarizing him with the language conveying the ideas, position, direction, 
ete.; and, second, of picturing out the various combinations with more dis- 
tinctness and freedom from distracting surroundings than can be attained 
in any other mode of illustration. 

The Diagrams, which, after Lesson V, take the place of the slates, are 
designed for more extended slate exercises, still involving the use of num- 
bers. Their usefulness as primary drawing-lessons can not fail to be rec- 
ognized. 

The children are not only to discover and describe the various combina- 
tions depicted, but, taking them as models, are to exercise their ingenuity 
in making as many others as possible with marks, dots, etc., upon their 
slates, or with sticks and other counters. ‘This is a training in form, num- 
ber, and arrangement, and, if counters of different colors can be obtained, 
admirable lessons in color may also be given, Thus is the inventive faculty 
called into play, imagination exercised, and taste cultivated, while the child 
is becoming accustomed to the number in all its combinations. The con- 
stant handling of the number, in this and the other exercises, affords the 
child experience, and, as far as the purely arithmetical aim of these first les- 
sons is concerned, this is the sole object, not to teach him to say that 2+ 
1+2=5, but to lead him to know it by experience. 

Figures and arithmetical signs have, of course, no place in Part I. 
Normal or schematic representations and names only are given. These 
are to be learned by sight, as suggested in the general notes. 

In the Seript Lesson at the foot of each page will be found the name 
of each number in script form. This is given for copy-work. 

Hints.—The suggestions and questions to be found under this heading 
on the teacher’s page are merely intended as hints of an extended work and 
great variety of exercises, to be supplied at the discretion of the teacher. 

The Parts from I to V indicate divisions of the subject; they do not 
correspond with successive grades in school classification. 


P fs gad b fie 


Suggestions for exercises to precede and supplement the /essons an 
the i/lustrated pages. 


SLATE EXERCISES. 


Object Lesson on S/ate,—As the slate and pencil are most important 
implements of study in this branch, it is well to make them first odjects of 
study—for one reason, that the child may become so well acquainted 
with them and their use that his ignorance of the tools may not interfere 
with the efficiency of the work to be done with them, and, also, because 
the study of them affords excellent opportunity for certain preparatory 
work necessary to the introduction of written arithmetic. Although every 
teacher has, doubtless, her form for object lessons on the slate, yet sugges- 
tions are here given for such lessons, that certain points having a more es- 
pecial bearing upon the subject under consideration may not be overlooked. 

The children are first encouraged to tell, in their own way, all they can 
see or be led to observe about their slates; then, by skillful questioning, 
not hinting too much as to answers expected, the teacher draws out the 
following, in consecutive form. /rame.—Use of frame; what made of; 
if covered, with what, and why [a little lesson on quietness]; if rubber 
corners, why. Jor slate part.—Of what made; hard; breakable; color. 
Parts.—Sides, how many; corners, how many; faces, one looking up 
at you, upper face; one looking at desk, lower face; how many faces 
has a slate? How many faces have you? Care of slate-—Breaking, scratch- 
ing, cleanliness, etc. 


An Object Lesson on the Pencil has its place here, and the following 
points to be made are suggested: A conversation on pencils in general; use 
lead-pencils for white paper, black on white, stone-pencils for slate, white 
on black; compare crayon and blackboard. Shape.—Long, round, pretty ; 
easy to hold. Hnds.—How many; one blunt, one sharp; why sharp. How 
to hold pencils.—Position of fingers; of wrist [writing position]. Use.— 
Make firm, light lines; heavy lines scratch and are not easily erased. Care 
of pencils.—Easily broken; carefully handled; kept sharp for neat work 
and light lines. 


6 SUGGESTIONS FOR EXERCISES. 


Lessons on Position and Direction.—It is an accepted theory that the 
child must go from the known to the unknown, and that each newly-ac- 
quired experience be made a stepping-stone to the next. Few children on 
first entering school, without previous training, would be able to obey the 
direction, ‘‘ Make a ring in the left upper corner of your slate.”’ On the 
other hand, few, if any, would be found that did not know which is the 
right band. With this bit of “terra firma” to stand upon, the teacher, 
after giving a short and entertaining exercise upon ‘‘right hand and left 
hand,” begins the lessons on the use of the slate by the following exercise, 
which has for its purpose the learning of 

Position.—The children being first directed to place their slates lengthwise 
on their desks {the word will be readily learned by ‘‘telling and doing ”’], 
the teacher begins by having them point out the right side of the slate; left 
side ; the lower side [side lower down on desk]; upper side [side farther wp 
on the desk]. ‘‘ How many corners on left side? Where are they? Who 
can point to a left corner? To the left wpper corner? To the left lower 
corner? Who can tell me the name of this corner [pointing to corner of 
slate pictured on blackboard]? Who can find the right upper corner? 
Who ean find the right dower corner? Jennie may point to any one corner 
of her slate and tell me its name. Johnnie may show and tell another, etc., 
etc.” After this follows an exercise on 

Direction.—‘‘ Put your finger on the left side of your slate, in the upper 
corner. Now slowly move it downward as far as you can. Where does it 
stop [left lower corner]? Where did you start? Now start at the left 
lower corner, and move your finger upward. Where does it stop?” Re- 
peat with the right side. ‘‘ Put your finger in the left upper corner again. 
Now move it slowly across the top of your slate, toward the right side. 
Where does it stop [right upper corner]? Where did you start? Now 
start at the right upper corner, and move it back toward the left.” Ques- 
tion as before, and then repeat with the lower side of the slate. 


CLASS EXERCISES, 


Tue Class Exercises outlined here are given only suggestively, to be 
amplified or condensed, omitted or repeated, at the discretion of the teacher ; 
their purpose being, mainly, to show what kind of oral work may advan- 
tageously supplement the work for the children’s pages. ~ 

By means of such exercises as the following, each of which has its definite 
purpose, it will be found that memory is cultivated, imagination stimulated, 


SUGGESTIONS FOR EXERCISES, — 7 


and observation trained; sight, hearing, and touch are exercised, and thus 
a harmonious development of the child’s powers is attained; the habit of 
attention, more or less sustained, is formed, and prompt action in obedience 
to command is acquired, as also a ready expression of thought. 

1. The teacher calls upon Johnnie to come and find out what she has in 
her right hand. Johnnie finds ‘‘a marble.” Jennie finds ‘‘two marbles” 
in her left hand. ‘‘ Would you rather have Jennie’s marbles or Johnnie’s? 
Why? Who would rather have mine [showing a handful]? Why?” 

2. ‘*How many more marbles has Jennie than Johnnie? What can I do 
so that Johnnie and Jennie will each have the same number of marbles? 

3. ‘Ned, bring me one little girl; now bring me one little girl again. 
How many times did Ned bring me one girl? [Putting arms around them 
and bringing them close together.] How many are there? [Sends one to 
her seat.| How many did I send back? How many are left here? 

4. “Hold up as many hands asI do. Hold up twice as many. Who 
can show me this many [two] pencils?) Show me half as many.” 

5. ** Now, let me see all the little heads bowed down upon the desks. 
Shut your eyes tight. Listen! [Claps twice.} Wakeup! Who can tell me 
what he heard? How many claps? 

6. ‘* Who would like to play blindman? Well, blindman, feel these 
pebbles and tell me how many there are.” She tests him with numbers, 
from one to three, and then with a large number, calling out the expression 
“many pebbles.” 

7. “Clap your hands this many times. [Makes two rings.] Clap once 
for each star I make. [Makes ten stars, and covers them quickly.| Tell me, 
without seeing, how many stars I made. How many times you clapped.” 

8. “Show me as many counters as you have heads; arms; chins; cheeks.” 

9. ““ Who knows of something that has one wheel? Two wheels? Three 
wheels? Two feet? Four feet? More than four feet?” 

10. ‘‘ How many eyes has a cat? What has one eye? [Needle.] How 
many ends has a pin? Namethem. How many wings has a bird? A fly? 

11. If Nellie earns one penny making lamp-lighters to-day and one 
penny to-morrow, how many will she have? If you had two pencils, and 
lost one yesterday and one to-day, how many would you have left?” 

12. ‘* Nellie, find two blue stars [paper]. Jennie, find one red, one blue, 
and one yellow star. Walter, find three different colored stars.” 

13. Who can touch two different things? Three different things? 

14. ‘tI hear the clock ticking, a bell ringing, and Will writing on his 
slate. How many soundsdol hear? Who can tell of two different sounds?” 


8 LESSONS ON ONE AND TWO. 


/, Outline of Lesson on the Words and the Schemas,— What is this, 
children [e] [pointing to it on the blackboard]?” ‘‘ A dot.” ‘‘ How many 
dots?” ‘ Onedot.” ‘‘Say again the word which tells how many dots; say 
it slowly. It sounds like the tone of a great bell. Would you like to see 
how it looks?” The teacher prints slowly, and in large letters, the word 
one, and then calls upon individuals to.point to it and pronounce it. Then, 
‘“Show me one book; one slate,” etc. ‘Show me [pointing to the word 
but not pronouncing it] [one] pencil; [one] desk,” ete. 

In the same way, by showing [eo el, the word “two” is introduced. 
‘‘Sounds like the cry of an owl.” The teacher then prints the word, as 
before, and lets the printed form ‘‘ tell” how many fingers, hands, books, 
etc., to be shown or pointed to. Changing the questioning from one thing 
to two things, she tests their conception of the meaning of the two words, 
Then, showing one and two of various articles. she calls upon first one and 
then another to point to the word which tells how many things she is show- 
ing. The class is then permitted to find these words (and show how 
many) in chart or readers, or any text at hand, and finally to discover both 
the words and the schemas in their arithmetics. 


/1, Hints for Language Lesson on“ The Conversation,’’—The general 
aim of this exercise is to give a liberal training in reading, and expressing 
the thought contained in pictures, while the special points to be brought out 
through this medium are a recognition of the number illustrated, first as a 
whole and then as to its groupings. 

This is to be attained by means of a familiar conversation in which the 
children are encouraged to give full and free expression to the ideas they 
gain from the picture, their attention being directed to special points by the 
questions of the teacher. Since the principal aim of this lesson is to stimu- 
late thought and cultivate the imagination, it would not be advisable to 
throw the child off the track of the thought by insisting upon accurate 
description, nor by too frequent corrections of language. 

The teacher would do well to first bring the children into sympathy with 
the beauty and spirit of the scene in this first illustration, by noting the 
softened light of early dawn, the fleecy clouds, the rising sun, the long 
shadows, and the signs of awakening life. ‘‘ How many of you ever saw 
the sun rise? Who ever saw the moon? The stars? How many stars? 
How many suns? What are the children doing? Do blackberries grow on 
trees? What is the man doing? Is this place in the city or country ? 
What else do you see in the picture?” This last will call out an enumeration 


LESSONS ON ONE AND TWO (continued). 9 


of the objects in the picture, from which, together with the above questions, 
the teacher may skillfully draw out a more or less connected narrative, 
which will be “the story the picture tells.’ The special point, number, 
may be further developed by questions such as the following: ‘* How many 
hands has the boy? How many has the girl? How many eyes has the 
boy? Do you see the girl’s eyes? . How many feet has the boy? How 
do you know? Choose the colors you would have for this little girl’s dress ; 
her apron, etc. How many hats has the girl? The boy? How many hats 
in the picture? Count the sheep. Count the birds. How many more 
doors than windows has the barn?” ete. 


//1, Slate-work,—In all Recitation Exercises upon the slate, let the con- 
scious effort and ambition of the children be “to tell a nice long story,” 
i. e., to give a full and exact description. The degree of accuracy with 
which the child describes the slate, and the location and arrangement of the 
objects thereon, will be a test of the exactness of his observation. With 
this in view, corrections of language can advisedly be made here, and the 
exercise may thus serve as a training, both in keenness of perception and 
accuracy of expression. 

In beginning the lesson on the slate in the book, the teacher should first 
call the attention of the children to the resemblance of this pictured slate to 
their own slates; also to the differences. ‘‘ How many corners has your slate? 
How many has this slate? Are they sharp corners or round corners? How 
many sides has this slate? Has yours? How many faces? And yours? 
Point to the left side of this slate; upper side; right side; lower side. 
Who can tell me what he sees in the middle of this slate?” Require a full 
and correct statement, first as to ‘‘ what” and ‘ where,” and, after arrange- 
ment has been discovered, as to ‘‘how” arranged. For example: ‘I see 
[or there are] two stars in the upper right corner of the slate. The stars 
are one under the other.” Or “I see two flags, side by side, in the lower 
left corner of the slate,” etc., etc. Every such statement will, of course, 
have to be at first built up, point by point, because the child is as yet un- 
trained in observation, and does not see it all atonce; and, also, because he 
has not learned how to express what he does see, After all the objects on 
the slate have been thus ‘‘ located,” comparison as to number is next noted. 
“How many flags are there? How many more flags than rings? How 
many stars? Are there more stars than rings?” etc., ete. 

A Drawing Exercise may follow, in which the children are directed to 
either copy on their own slates the objects as they are on the pictured slate, 


10 LESSONS ON ONE AND TWO (continued). 


in their books, or on a similar one which the teacher shall have drawn upon 
the blackboard. 

The Dictation Exercises should be introduced by a more or less rapid 
review of the object-lesson upon the slate. The little workers being ready, 
with slates in proper position, the teacher directs them to make “‘ a row 
of stars down the left side of the slate; across the upper side; down right 
side; across lower side.” Then, on the other face, to ‘‘ make a star in the 
upper left corner; two rings, side by side, in the lower right corner ; asquare 
in the center of the slate,” etc., till each place is located and filled. 

/V, What can you te// ?7—In these exercises the children are to be left, 
as much as possible, first, to give spontaneous and unaided expression to 
‘the story the picture tells’; and, second, to observe the detail with special 
reference to number. The following are some of the points that may be 
made in these pictures : 

First picture.—A rabbit in field; standing on two hind-legs; holding up 
two fore-legs; has two long ears; two eyes; we see only one eye, etc. The 
lily has one open blossom, two buds; half-way up there is only one stem; 
above, two stems; two leaves on stem with one blossom on it; one leaf 
on the stem that has two buds, etc. By counting, we find there are ten 
leaves in all. Second picture.—A garden; wall, with bicycle against it; and 
a wheelbarrow; vine on wall, etc. Bicycle has two wheels; the wheel- 
barrow has one wheel (compare); bicycle has two pedals, two handles, one 
saddle, etc.; wheelbarrow has two handles, two legs, ete. Third picture.— 
Looking out of the window; a bird-cage; bird flying away; no bird in 
cage now. ‘How do you think it happened that the bird got out?” Out- 
side the window a house can be seen, etc. Fourth picture.-—Boys flying 
kites in a field or vacant lot. Two boys, each boy one kite; so two kites— 
twisted together; each kite one tail; two kites, two tails; same with 
strings. If one boy runs off with his kite, one boy and one kite left, etc. 

V, The Seript.—As each school has its established system of writing-les- 
sons, the following brief suggestions are only offered: That the recognition 
of the written form of the word may be given in the same way as the 
printed form; that the written and printed forms be compared; that the 
written word on the blackboard be traced with the pointer, individually, and 
traced in the air with pencils by the class; and, finally, that it be copied, 
first as a whole, and then practice given on the accompanying letters. 

V/, Hints,—The oral work on One and Two will be found on page 7. 
For ‘‘ busy work ” hints, see lessons on Three. 


The Conversation, 


12 LESSONS ON THREE. 


/, The Word and Schema.—As in Lessons on One and Two. 


/!, The Conversation.—After the general conversation the following 
special points to be made: ‘‘How many little girls are there? Names? 
Each girl has one apple; doth want the other one. If (May) takes it, how 
many will she have? How many more than (Nita)? Which will have twice 
as many as the other? Which half as many? What can be done so that 
each will have as much as the other? Look at the picture below and tell.” 

In the second picture the story will be readily grasped by the children. 
A few questions only will be needed to give direction to the thought, 
and bring out the facts in the number. 


//!, Slate-Work,—Recitation: The pupils to give a full and exact de- 
scription of each group, as ‘‘ what,” ‘‘ where,” ‘how many,” and “how 
arranged,” as ‘In the middle of the lower side of this slate there are three 
flags, two of the flags are side by side, and the other flag is below them.” 
Also compare with slate in Lessons on One and Two. 

Drawing: The children may be directed to copy exactly what is on the 
pictured slate, or, to devise original arrangements of these same objects. 

Dictation: Review as in the first lesson. Then dictate from the pictured 
slate; second, have the groups placed in the corners; and; third, dictate 
ones and twos of objects to be drawn, as well-as threes. In each case 
have the written work described by individuals. 


/V. What can you te/l/?—The story of the hen and her three ducklings 
will be easily gathered by the children from the two pictures. The points 
in number to be noted are: There are three ducklings and two water-rats. 
If two rats get one duck each, then two will be taken and one left, etc., etc. 


V. The Script.—To be taught as directed in Lessons on One and Two. 
V/, Hints for busy work, 
to be copied from the black- [e e| [ol © © © Ad 
board by the pupils. [e @ e| 
Rea A A A 


Have the children write 
or tell number stories about e e eee ies 
e ee 24 


objects in view, as, ‘‘There 
are (2 books) on the (table) 
and (1 book) on the (chair). Objects to be represented in outline drawings. 
“Draw a picture of three things, and write or tell a number story about it; 
mention three red things seen on the way to school (other colors); three 
things with wheels; three like things; three different things.” 


Oo”, 


? 


The Conversati 


What can you te// 


14. LESSONS ON FOUR. 


/, The Word and Schema.—To be given as suggested in preceding lessons. 


/1, The Conversation.—After the general conversation upon ‘‘ The Nut- 
ting Party,” the following points in number may be noted: 1 child in a 
. tree, 3 on the ground—4 in all. 2 children with hats off, 2 with hats on— 
4 hats in all. 1 child kneeling, 2 standing, 1 sitting in tree—4 in all, ete. 
‘‘What do you think the boy in the center picture is going to do? What 
hashein hishand?” Point to 2 things alike in this picture. Find 3 different 
things. How many pints of nuts were there in the basket? If he sells 2 
pints at 2 cents each, how much money will he get for them? 

///, Slate-Work.—fecitation: As in the preceding lessons. Special 
questions as to number may be, ‘‘ How many dots in the center? If you 
erased 1 star, how many would be left? How many times could you erase 2 
squares? To how many boys could you give 1 flag each?” ete. 

Drawing: Besides the exercises in the preceding lessons, the class may 
draw picture-slates of the size of that in the book, and exercise their ingenu- 
ity in varying the location and arrangement of the pictured objects. A 
training-lesson in language may be founded upon this exercise, by having 
each pupil describe in precise terms one of his picture-slates. 

Dictation: These lessons would differ from the preceding ones only in 
the greater variety of exercises possible from the material suggested. 

lV, What can you te// ?—The exciting incident pictured in this series 
of three pictures can not but unloose the tongue of even the shyest child, 
and ‘“‘ what they can tell” will find spontaneous expression, which need only 
to be directed. Compare 3 cats, 4 mice. ‘‘How many mice get away? 
Are caught? How many does each cat get? How many fore-feet has a 
cat? Hind-feet? How many more tails have 4 mice than 3 cats?” ete. 
In the picture, ‘‘ The Horse at the Blacksmith’s Shop,” the points in question 
are: ‘‘How many legs has a horse? A boy? How many more has the 
horse than the boy? How many times 2 shoes does a horse wear? A 
boy? Which has twice as many feet as the other? Half as many?” ete. 


V, The Seript-Lessons.—To be given as suggested in lesson on One and 
Two. 

V/, Hints for busy work, see ‘‘Three.” Have the class o 2 2 ®Q 
make a picture-story of a boy who had 4 cents, and bought 
3 sticks of candy at a cent apiece, as shown here. “If I | | | 
give Carl one fourth of these four pencils, how many will he get? How 
many shall I have left? How can I give these 2 apples to four boys? 
Would you rather have these (4 pennies) or these (2 two-cent pieces)?” 


The Conversation, 


- ———— SES 
= Se SE 


16 LESSONS ON FIVE. 


/, The Word and Schema.—To be given as suggested in Lessons on Three. 


/1, The Conversation.—‘ Tell me a story of your own about this tea- 
party ’’ will call out the individual ideas gained from the picture. ‘ How 
many children are talking? Listening? At the sides of the table? How 
many cups and saucers? Blocks? If each child takes a block, how many 
will be taken? How many left? etc., ete. 

‘‘ How many leaves on each stem of the ivy? How many ‘fives’? 
Count the ivy-leaves by ones. How many leaves on each stem in the rose- 
vine? How many blossoms? How many petals in each blossom? I see 
some rose-petals falling. Who can tell a number-story about this?” 


///, Slate-Work.—Recitation: After each group of pictured objects 
has been fully discussed and all the combinations within five found, com- 
pare with the slate for Four to find resemblances; and with that for Three 
to find differences. Then compare the groups, as the flags with the flags 
on all the preceding slates, as to number, location, and arrangement. 

Drawing: The suggestions given on all the preceding lessons may be 
put into use here, especially the last one on Four. 

Dictation: ** Quick-work” exercises in locating and arranging at once 
and then describing exactly and fully the matter dictated may serve both 
as a review and as a test of the child’s understanding of the terms he has 
been accumulating in the preceding lessons. 


/V. What can you tel/ ?7—“ Study the picture, ‘The Dog-Show,’ and tell 
me a story about it.” Direct the thought by questioning as to the kinds of 
dogs, the number, etce.: “If the Newfoundland dog should pick up the 
poodle and carry it off, how many dogs would be left? Suppose a rat 
should go scampering past, how many dogs would be left? How many tails 
do you see? How many don’t you see? Tell a number-story about each 
dog.” The next illustration was designed for the comparison of “ five” 
with numbers under five, and also for a lesson in comparing the feet of 
these animals with the human hand. The toads and toadstools illustrate 
two twos in five, and one left over, etc. “If that little toad goes off, how 
many will be left? If the wind blows down 1 toadstool, what then?” etc. 


V. The Script.—To be given as heretofore directed, and a comparison 
made between this word and the preceding ones. 

V/, Hints;—Draw your hand. “If I had 5 cakes, to how many little 
girls could I give 2 cakes each? John has (showing 5 pennies), and Carl 
has (showing a nickel), which can buy the most candy ?” 


18 LESSONS ON SIX. 


/, The Word and Schema.—To be given as heretofore suggested. 


//, The Conversation.—“ Coasting” and ‘‘ The Snow Fort” will require 
but little, if any, questioning to draw out “the story the picture tells” of 
these the favorite winter sports of childhood. The topic, ‘ Seasons,” 
might also be introduced here. 

Special points: First picture—‘* How many sleds? How many boys on 
the first sled going down? On the second? Altogether? How many girls 
going down?” ete., etc. Second picture—‘‘ How many boys in the fort? 
In front of the fort?” Other things to be examined as to number: snow- 
balls, trees, children, birds, and branches of trees outside, etc. 


///, The Diagrams are, first, to be examined by the pupils to find the 
combinations in six, and read in class; second, to be reproduced at the 
desks, either in blocks or in squares of various colored pasteboard, or copied 
ontheslate; third, to serve as models from which the children are to make 
other and original designs. The inventive faculty will thus be exercised, 
and training almost without limit, in form, arrangement, and combinations 
of the number afforded, as also invaluable lessons in color. 


lV, What can you te// 7—First picture: ‘‘The mother has four children 
and six cakes. How can she divide the cakes equally, and how much will 
each child receive?”? Other points are, the comparison of other numbers 
under six. Second picture, the story of ‘‘ The donkey that wouldn’t go”; 
time of day and season. Special points, ‘‘ How many children in the cart? 
Out of the cart? Altogether? On front seat? On back seat? How many 
feet has the donkey? The boy? Double as many? Half as many? Both 
together?” ete. Third picture, ‘‘The Rainy Day.” Make up a little story 
of your own about this picture. Special points, ‘‘How many little girls 
are there under this umbrella? How do you know? How many pairs of 
arms? Eyes? Heads?” ete. 


V, The Sor/pt.—Lessons to be given on the plan of the others. 


V/, Hints,— Draw as many rubber boots as three boys would wear. 
As many mittens. How many more legs has a spider than an elephant? 
Than a fish?” ete. A mother said, “I will give you one half of these six 
cakes if you tell me how many that will be?” ‘Draw six cents on your 
slate. Draw as many pencils as you can buy at two cents each.” Have 
the children make six-inch rules—of paper, pasteboard, or wood—for them- 
selves, and encourage them to measure their desks, slates, books, etc. 


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20 LESSONS ON SEVEN; 


/, The Word and Schema.—As previously suggested. 


//, The Conversation.—Upon wild rabbits, pet rabbits, pets in general. 
Characteristics of rabbits. ‘‘Cousin Hare”; fable of ‘‘Hare and Tor- 
toise,”’ story of ‘‘Brer Rabbit and the Tar-Baby,” from “‘ Uncle Remus,” 
adapted by the teacher, are suggested as topics for the general language- 
lesson. Special points in number upon all the combinations in seven may 
be made in the groupings of the rabbits. 


///, The Diagrams may be copied in lines on the slate or represented in 
objects by match-sticks, splints, etc., and, besides the simple arrangements 
showing the combinations of number, may represent real objects, such as 
houses, fences, chairs, tables, flower-pot with plant, and an endless number 
of like things, and also fancy geometrical designs and figures. 


lV, What can you te//?—The merry circle around the Christmas-tree 
will arouse the pleasantest recollections, and may be turned to great ad- 
vantage in inducing the children to relate their own experiences. Hereto- 
fore the work in language has been either to describe or narrate. After the 
general conversation, each child may be called upon to relate an experience ~ 
which shall have in its plot some combination of seven. The second pict- 
ure was specially designed to give occasion for a language-object upon “the 
table,” manners, setting a table, etc., while affording excellent material for 
the study of seven. ‘Tell all you can about the family who are going to 
have tea. How many grown people? Children? What ages? Where 
going to sit?’ etc. Special points also on the number of things on the 
table. The third picture shows an incident familiar and easily described. 
All the combinations of seven may be noted in the balloons. 


V, The Script.—As in previous lessons. 


V/, Hints.— Put seven blocks on your desk. Find another number in 
these blocks—another.” (Six, five, four, three, two two’s, and two three’s 
will be found.) Have a pupil tell a number-story, and the class picture it. 
As “Seven birds were on a fence, and three flew away.” Write ‘“‘ Four 
and three are ——,”’ and have the children copy, fill blanks, 
and complete. ‘‘ Write a number-story of your own about the Christmas- 
tree.” ‘‘How can I divide seven oranges equally between two children? 
Equally among three? Who can picture it? How many 2-cent stamps can 
I buy for seven cents?’ The children should have breakable objects, and 
be directed to find one half and one third, also to put together and compare 
with an undivided whole. 


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22, LESSONS ON EIGHT. 


/, The Word and Schema.—As heretofore, the word as the spoken or 
written, and the schema as the concrete expression of the number. 


/1, The Conversation.—“ The Apple-Gathering ” affords an effective 
lesson on country life and occupations. The topic of seasons may be re- 
sumed, the teacher taking the children back in memory to the spring-time 
when these trees were white with fragrant blossoms that spring rains 
and summer suns have ripened to the round red fruit. Now, in the 
autumn, the father and his children gather the apples, and early in the 
morning he goes off to the city and sells them. Here, also, we see the fruit- 
store of the man who bought the apples. ‘What is he doing? What 
measure has he in his hand?” Special points, the grouping of the barrels, 
the persons, baskets, trees, horses, ete., and the articles in the store. 


///, The Diagrams are to be used in the same way as those for six. 
They may be represented in objects by pennies, button-molds, etc., and by 
variously colored pasteboard disks. The forms may each be read in several 
different ways, as for instance: No. 4 may be read as either 3 and 2 and 3, or 
as 1 and 8 and 38 and 1. ‘The class may also be asked to ‘find other num- 
bers” in the eight. One may find a three, pointing to the upper row in 
No.1. The question, ‘‘ What eise is there besides the three?” will call out 
the fact that 3+?=8. In No. 2, one may find “a four,” and immediately 
discovers that there are ‘‘two fours” in the eight spots, ete. 


/V, What can you te// ?—The children may “imagine” or ‘ dream” 
stories about the people to whom this room belongs, two number combina- 
tions of eight being required as the basis of each story. Special points, the 
panes of glass, legs of tables, chair, etc. In the second picture there will be 
no lack of “stories,” which may be given direction by requiring, as before, 
number to be the basis of them. The third picture is presented for the 
comparison of the even numbers under eight with eight. ‘‘ How many more 
blossoms would you have to add to the first plant to have eight? Which 
plant has twice as many flowers as another? Three times as many? How 
many times could you pick two of the lilies?” etc. The comparisons are 
numerous; only a few types have been given. 

V. The Script.—To be given as previously suggested. 

V/, Hints.—Name eight different things that can run, hop—that have 
eyes, ears, hair, fur, wings. Eight kinds of fruit, vegetables, nuts, games. 

Before the eyes of the children, the teacher folds a paper in four folds, 
and then cuts out two paper dolls, and asks the children to guess how many. 


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The Conversation, 


24: LESSONS ON NINE. 


/, The Word and Schema,.—To be given as heretofore suggested. 


/1, The Conversation,—This beautiful sea-shore scene will afford so 
many and varied object language-lessons that only a few hints on each of 
the different lines of thought suggested by it can be given: Land and water 
—the sea-shore—rocky coast—sandy beach—calm and storm—light-houses 
—pleasures of the sea—toils of the sea (fishing, whaling, transportation)— 
ships and boats—the different kinds of motive power—the vessels in the 
picture—what the sea yields for our use—things under the water—at the 
bottom of the sea. These last two are illustrated by the frame of the pict- 
ure, and give opportunity for opening a new world to many of the children. 
Interesting facts about the wonders and beauties of submarine life may 
be gathered from almost any natural history. The special points in num- 
ber may be brought out by questioning on the ship-sails; the legs, ete., of 
the crab; the points of the star-fish, the sea-anemone, etc., ete. 


//!, The Diagrams,—Too much stress can not be laid upon the value 
and variety of the exercises which may be founded upon these diagrams. 
The designs which may be formed, either with long counters or by lines on 
the slate, are almost without limit, and, if the pupils are required to pay 
attention to the combinations of number in making their designs, these ex- 
ercises can not fail to familiarize them with the number in its every aspect- 


lV, What can you te//?—After the story of Little Bo-peep has beer 
rehearsed, require each pupil to tell in what groups he thinks the sheep ran 
off: one will say, ‘‘ First three, and then three more, and then three more,” 
and so on, with all the combinations in the number. Again, looking at the 
illustrations from different points of view, many of the combinations can 
be seen, as, one and eight, or one and three and five, etc. The story and 
illustration of ‘‘ Little Boy Blue” may be treated in a similar way. 


V, The Script.—This lesson introduces a new feature, that of the slant- 
ing lines, which serve for practice in slant and distance. 


V/, Hints,—Draw a number-story picture of nine boys and six sleds. 
Draw another of three rabbits and nine carrots. Draw nine oblique lines 
(in groups of three, or five and four, etc.). Draw nine horizontal lines; 
nine vertical lines. Have exercises in making up nine with one, two, three, 
and five cent pieces. Conduct sales of slate, pencils, sponges, paper-dolls, 
tops, etc., and have the children picture the operations (as in ‘ Hints,” 
Lessons on Four). Have the children each bring a flower, and find and 
write number-stories about the parts of the flower. 


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The Conversation, 


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26 LESSONS ON TEN. 


!, The Word and Schema.—As heretofore. 


/1, The Conversation.—The game ‘ Keeping Store,”’ besides being of un- 
failing interest to the children, gives special advantages for the study of the 
combinations within ten, as in buying and selling, exchanging one number 
of things (pennies) for another, and ‘‘ making change,” etc., both customer 
and merchant must be ready with his calculations. A study of the articles 
for sale will also yield most of the combinations. Each pupil may be 
allowed to make an imaginary sale or purchase of the things pictured here, 
and tell of the transaction. 


//1, The Diagrams in this lesson are like those in six, and may be 
used in the same ways. With space less limited far more beautiful designs 
may be made. It is specially desirable that the pupil should not, in these 
exercises, be permitted to overlook the element of number, -but should 
note the combinations he forms. In this, and in the other exercises founded 
upon the diagrams, may be produced what Froebel calls ‘forms of life— 
such as actually exist, and come under our observation as works of art and 
industry ; forms of knowledge—such as relate to number, order, proportion, 
etc.; and forms of beauty—representing ideal forms, models of symmetry 
and order.” 


/V. What can you te//?—If the children have been duly encouraged in 
their previous work to give a numerical turn to the stories they tell, the 
teacher will find no difficulty in gathering from the various versions which 
they will give of ‘The old woman who had so many children,” etc., and 
‘St. Nicholas,” the many combinations within ten—combinations which 
will cover the ground of the four rules of addition, multiplication, subtrac- 
tion, and division, without, however, having these recognized by the chil- 
dren as distinct operations. 


V, The Script consists of a lesson on “ ten,” with a review of ‘‘eight” 
and “nine.’’ A review should also be made of all the previous script-lessons. 


V/, Hints.—Make a paper disk and perform operations in fractions be- 
fore the class, and have them tell what you have done. Then let each child 
make at home a “ paper cake,” and then have them divide their “ cakes” in 
halves, fourths, eighths. This will afford many interesting exercises. At 
another time the cakes may be divided into thirds and sixths. By compari- 
son, and by putting parts together, without formulating what they do, they 
will be unconsciously learning to perform with parts all the operations that 
they do with wholes. 


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28 NOTES ON LESSONS IN PARTS II, Ill, AND IV. 


Part //,—* Lessons I to X. The main purpose of these lessons is to 
introduce the child to the printed forms of arithmetical expressions. They 
do not embrace all, nor the majority, of the combinations within ten, but 
may be taken as types for blackboard exercises, in which are introduced 
the other combinations. They constitute a systematic series of reading- 
lessons, and may besides be copied on the slate, the children filling the blanks 
with pictures of the objects, and the names of the numbers, thus serving as 
writing and drawing lessons also. 

* Lessons VI, VII, and VIIJ.—‘“ Making up and writing original prob- 
lems.” These illustrations are given as themes for oral, blackboard, and 
slate exercises, in making original number-stories like those in the text. 

* Lessons XI and XII. Ten is here dealt with as a unit, that the child 
may get, at the outset, a correct view of our decimal notation. 

* Lesson XIII. The first of a séries of test-lessons, and which are, there- 
fore, purposely miscellaneous in character. 

* Lesson XV. The purpose of this lesson is to picture out the real mean- 
ing of “eleven, twelve,” etc., and make the child realize it when counting. 

* Lessons XX to XXIII, and also Lesson XXXV. A summary and re- 
view of the numbers within ten, analyzed according to the Grube method. 

* Lessons XXXII and XXXIII may be either dictated by the teacher, or 
used as a silent desk exercise. 


Part ///,—* Lessons IV, VIII, XV, XIX, XXII, and XXV, constitute 
the groundwork of all operations within the hundred; each one introduces 
a new step, and should be thoroughly practiced with objects. 

*Lessons XXIV to XXXII embrace a complete review of numbers 
from ten to twenty, which should now be thoroughly practiced. 


Part [V.—* Lesson IV. In all cases of successive additions, subtrac- 
tions, etc., state results, not the operations that produce them. 

*Lessons V, VII, 1X, XIJ, XVI, XXII, XXIX, and XXX, each intro- 
duces a new step, and it is especially recommended that objects be used for 
illustration until the pupils have thoroughly mastered the subject. 

*Lesson XV. In this, as in all the lessons on fractions, pupils should 
be required to draw squares, and actually perform the operations with them 
before expressing in figures. . 

* Lessons XX and XXIV. The large numbers given here are for prac- - 
tice exercises only—concrete applications in them would be inadvisable. 

* Lesson XXV. It is recommended that the teacher herself illustrate 
each step of this lesson with objects. 


PAR TELE: 


READING LESSONS IN NUMBER, 
AND DICTATION EXERCISES. 


Combinations from One to Ten, 


LESSON I. 


ADDITION, 


Phrase and words to be /earned by sight. 


| How many | and = are 


One —— and one 


2. How many, are aye) & 


Two ——— and one 


. How many , are 
One —— and ‘one ———. and one 


4. , How many, are 


eB ON ee BN ONE cee cS 0 eee 


380 LESSON II. 


Phrases and words to be learned by sight, 


EL See | a in on 
There are | 


Two. and two 2g Pp ae 
SE ZF 


ope 


many in all)? 


Two —— = 4nd one Send two 2 Sere 


LESSON III. 31 


Words to be /earned by sight and sound. 


by tin’der ind arg 


How many In all? 


the 4. How many ,in all? 
Three ied PAHs reS: a GPAh As seat 


3. ,L see, eure’ in the <2 TEAS 
the By and OL, under the J 


How many in all e 


Two and two. end two are —+ => — 


32 LESSON IV. 


SUBTRACTION, 


Phrases to be learned by sight and words by sound. 


are left i 7 léss 
will be left , thése 


take away | fly away , hop away , 


1. If I take away, two of these SQ igs 


_ how many , will be left ? 

[eS ae es ee 

2. If two of these My f; a 
will be left? 


are 


fly away, how many 


Four less two are 


= — of these. ea. How 
many ,are left, ? 
Segoe 


4. If two of these 


hop away, and 


two | fly away, how many , will be left? 


is SS ae eT eee 


Five __— less two -_— less two 


LESSON. V. 33 


Phrases to be learned by sight and words by sound, 


, were there | left ran 
ran away | lambs  on’ly 


ran away, and_ these 


4 are left. How many , were there 


ts 
gall 


in all ? 


—__— lambs less three lambs are two lambs. 


ix There were, Sess Now there are | 


only two. How many ,ran away |? 


Six. /ess ares tWost- 


Mies. If three fly away ,, 


es 
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Wii 


how many will be left |? 


Six less three ——— are : 


4. If I take away, two of these 


and you take away, two, how many 


: will be left : ? 


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34 LESSON VI. 

Phrases to be learned by sight and words by sound, 
Here are, roll away | frogs 
of them, | swim away , bees 
out of runs away | goat 
1, | Here are, If one goat 


Tuns away ), oer many , will be left, ? 
Six goats less one goat are —— goats. 


If two of these run away, and 


you take away one, how many 


will be left, ? 


Six ——— less two ——— /ess one gre Lad be 


If four of them 


‘There are, <4 


hop away ,, how many | will be left ? 
Six frogs /ess ee frogs are 2 frozs. 


Here are, g If two ,run away , and 


three , swim away ,, how many will be left ? 


Seven less twoe.2-Yess thréess 229 ere ae, 


LESSON VI (continued). 35 


G25. If two bees 
See 


fly away ,, how many will be left |? 


Or 


. Eight bees are on this @ 


Eight bees less two bees are —— bees. 
. If you take ALLEL out of, this 


how many will be left? 


Pr) 


EI Rat oer Oe8 Ele (eae OSG ae, 


val —<— 


eee, )SWIM away |, 
how many will be left? 


~I 


are ——— ——, 


Nine ——— less six 


If five of them roll away |, 
how many will be left ? 


iepes = (ess five #3 are __—, 


* Making up and writing out original problems, 


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35 LESSON VII. 


MULTIPLICATION AND DIVISION. 
Phrases to be learned by sight and words by sound, 


How many times , légs this 
Are any left, fone DOX@ win ke 
can you find | tops kites 


1. How many legs have these ..44—f}—_: | 


Two times two legs are —— legs. 


: a, ts i He f 
2. | How many times ye mi have these bad tame 


How many in all? 
Three times !¥v ie OY Aoi tii a Meat 


3. , How many times, can you ,take away, two 


tops from these PM? How many 
= A At 


will be left? 


From five tops you can take, 
2) op el be Er 
4. In these OeKrgrtmmre 


times, 


Megas [A 
two ie "ca you at 


[n_ six. kites there are ——— times 


LESSON VII (continued). 37 


5. From these ¢ bow many times, can you 


take two pinks ! ? Are any left? 


From seven pinks \ you can take, 7 ist WANES, 
and —— pink will be left. 


6. How many times four legs have }y 


How many legs in all? 
Two times —— legs are —— legs. 
7. How many times [LES in this 
How many in all? 


There are times Bis inthe’ Dox, 


TINGR LG tee Oe ae ae, 


Making up and writing out original problems. 


38 LESSON’ VITE 


Words to be learned by sight and by sound, 


her apples from bids 
toes _ fingers éges nuts 


1. How many times can youtake TELE out of : 


this box ee Are any left? 
= ( - =e 
From six —— you can take Ga Ay A 


times. 


a Nis 
2. |How many times | five fingers on these a at 
How many fingers in all? 
[here sare == Vingers@ iN alle Ge) free a ers 


there are —— times five fingers. 


3. How many toes has a cat on her ¢@¥ | 


Two times five toes are ——— toes. 


4. How many toes has a cat on her 


Two. times —— toes are ——— toes. 


From eight buds you can take 


Ce all 


_ 


me | 


LESSON VIII (continued). 39 


How many times, three nuts are there on - 


How many in all? 
There are —_— times three nuts. Three times 2s 


are ——— «nuts. 


How many times | can you take four apples 


= 


4O *LESSON IX. 


Phrases to be learned. by sight and words by sound. 


one half, one fourth | mine 
one third, | one fifth, shélls 
as many | she gave | eatch 


1. Mark has ae®, but I have only , one half, 


as many, How many have I? 


One half of two — Is 


Vi. thoi! +n ¥Z 
y | OO pa but Carl has only ; one third | 
as many, How many has Carl? 
Onesthirdrofe tire? === 5a 


Liylhy, Wey — 
4 PP ai 


3. What is one fourth , of IS 


One fourth of four apples 1s —— . 
4. If you catch one fifth of these Jy 


many | will you have ;? 
One fifth of five —— is 


5. What is | one half, half, of eo.cy 9 Hip D 


One half of six is —— — 


reat te ee 


vey 


a 


ee a 


—- 


LESSON 1X (continued). 41 
6. May had ae e@ey She gave, one third, of 
them to Ned. a oh did he get)? 
One third of six —— are —— —— 
7. If I give one half, of these Reo you, how 
many , will you ee e)? 
One half of four —— SIE ee 
8. If I take away one half, of these Ee 
how many , will be left, ? 
One half of eight are so 
9. One fourth , of these Geax SS 
How many shells have |? 
One fourth ill eight shells are —— shells, 
10. Ann had ¢@ Nt ig ui She gave, one third 
of paid Tian me. How many ,did I get,? 
One third-of, nine: 2+ —. ace. = ex 
11. If you take , one half, of these < - “ie 
how many eggs , will you have ;? . 
One half of ten eggs are —— eggs. 


ee 


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42 LESSON X. 


Phrases to be /earned by sight and words by sound. 


more than, have l, log pond 


1. Carl has 


and I have 2. How 


many more has Carl than |? 


lhree —— are. —— more thanz one == 


2. Nellie has 4@ 


many more have | than Nellie? 


our eae hg —— ,;more than, one ——. 
3. May has ¥) and Ned has Ge. How 
many more has May than Ned? 
Five are 232 = 7 more: Wan oe 
4. Here are GED “2 In the pond there 


are five frogs. How many more frogs are 


there on the log, than in the pond ? 


Six frogs LE Sa more than five frogs. 
5. 1 have mye 


How many Ari Hb! have [, Te May? 


Eight ——*are —— ,;more*than, two ——. 


*LESSON XI. 43 


COUNTING BY TENS. 


One ten Two tens Three tens Four tens 
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ten. twenty. thirty. 

Five tens Six tens 


seventy. 


Ten tens 


eighty. aero one hundred. 


* LESSONCAIE 


DICTATION EXERCISES, WORKING WITH TENS. 
Phrases to be learned by sight and words by sound. 


| Give the name, like these , add make 


/have you made, ;enough more, e-rase’ made 


. Make two tens like these and name them. 
. Make four tens. Give the name : 


. Make three tens. Name them. Add 


enough more tens , to make, five tens. 


. Make six tens. Name them. Erase three 
tens. ‘Name the tens , you have left. 

. Make three tens and four more tens. How 
many tens in all? Give the name. 

. Make eight tens. Name them. Erase half 
of the eight tens. How many are left? 

. Make five tens. Namethem. Add enough 
more tens to have nine tens. 


. Make nine tens. Name them. How many 


times can you take, three tens from ninety? 


LESSON XII (continued). AS 


9. Make five tens. Name them. Add five 
more tens. How many tens in all have you? 


How many times, have you made five tens? 


: Give the name : of two times five tens. 


10. Make three tens. Name them. Add three 
more tens. How many tens have you 
now? Name? Add three more tens. How 
many times ; have you made, three tens 
(thirty)? How many tens have you in 

all? Name? | 
11. Make two tens. Name them. Add two more 
tens (twenty). How many tens have you? 
Name? Add two more tens. How many 
tens have you now? Name? Add two 
more tens. How many now? Name? 
Add two more tens. How many times 


,have you made, two tens? How many 


tens in all? Name? 


A6 *LESSON XIII. 


MISCELLANEOUS EXAMPLES. 
Phrases to be learned by sight and words by sound, 


Picture it has he left , shall — séll 
with counters, on your slate | slate dots 


1. How many are WA and Ave and we ? 
2. Ned had GQ : 


Fae 


and EHP m&. Four of 

them ran away. How many ,has he left |? 
; rs Axo, 

3. I have two times #4 and ade more. 
How many, have I in all? | Picture it, 
with counters, or with dots ,on your slate . 

€@)K& and two times Ber 

How many in all? | Picture it. 2 
. How many times >< in this 


Oy Haye ab Rs 


4. Ihave three times 


_ 8. How many are three times three kites? 


LESSON XIV. A'7 


Phrases to be learned by sight and words by sound, 


the more, to which | much  cénts 
the more money , more apples 


bo 


. Carl has 


_ Nellie found @ 


How many more legs has a ee. than an 


and May has 


G, : 
223. Which has 


the more money ,? 
Aa a eee ee ee 


. Nita has (8). 


she than Carl? How much more than May? 


How many more legs has a “than a SS ? 
4 % Tey a Cot <e 
fe ) 3 ; spy “Pm to 


Mary. ,To which did I give , the more? 


ess and Jack found 


only three shells) How many more shells 
has Nellie than Jack ? 
I have seven cents and Jamie has nine cents. 


Which of us, has , the more money 


much more ? 


How 


48 LESSON XV. 


COUNTING BY ONES, ABOVE TEN. 


One ten One ten One ten One ten 
and and and and 
one two three four 

e@ 
& @ 
6 e @ 

8 |e & e 
eleven. twelve. thirteen. fourteen. 
One ten One ten One ten One ten 

and and and and 

five IS1X seven ecight 
& @ ' 
@ - @ @ 

2) t a a 

e e s e 

Se @ @ & 

@ © e e 

fy e@ ®& e 
fifteen. sixteen. seventeen. eighteen. 
One ten Two tens Two tens Three tens 

and and eland 

. nine SIX five 

& 

e e@ 

a Pi) e 

e e e@ 

r e 2 

e e é@ 

@ <= eo” S 


nineteen. twenty. twenty-six. thirty-five. 


LESSON XV Kecaeinued ). AQ 


Four tens Five tens Six tens 
and 
four 


and 


one 


& 
oO 
e 
® ee e\e 
forty-four. fifty- two sixty-one. 
Seven tens and Hight tens and 
*lseven ee 
e ee 
° ee 
ele ee 
ele ee 
\@|@ @|@ 
ele @ @ 
ele ee 
ele ee 
@|e @\@ 


seventy-seven. ighty-three, 


ie 


Nine tens and nine 


ninety-nine. 


OS 


OS 


O LESSON XVI. 


GATHERING INTO TENS. 
The pupils to be directed to copy these groups of dots, putting the 
tens into boxes, and to te// or write in words how many tens 
and how many single dots over in each group, 
and to give the name of the number, 


The pupils to be directed to make as many dots as these numbers 
call for, and to put the tens into boxes, 


thirteen 
thirty-one 


fifty-seven 


fifteen 
fifty-eight 
seventy-two 
ninety-nine 


forty-six 


nineteen 
ninety 
sixty-three 
seventy-five 


eighty-two 


pv 


LESSON XVII. 51 


Words to be /earned by sound. 


twice found dais‘les dan’de-li-ons 


a Ma: Ae BaF to Tne and I he 


ae many had | in all? 


Ned has, half as many \. 


How many has Ned? How many have 
’ 
they in all? Picture it. 
5. If you give me_ half as many, daisies as are 


& » 
in this $¢: , how many shall I have? 


6. If you give me twice as many | daisies as are 


in this a how many shall | have? 


UNIVERSITY OF 
ILLINOIS LIBRARY 


52 LESSON XVIII. 


Phrases to be learned by sight and words by sound. 


Put into twice as many. bunch 
take out of half of the rest wheels 


1. Will gave these ea@jp to me. He gave 


twice as many, to Carl. How many shells 


did he give to Carl? , Picture it,. 
bes oes 
2. Rose had #2 


half of the rest. How many _hasshe left ? 


3 Se 


. She lost two and gave me 


He ate one bunch and 


gave one third of the rest, to Ned. How 


many , has he left,? Picture it. 


a <—, 
AS can you take out of, 


a 


LESSON XIX. 53 


Phrases to be learned by sight and words by sound. 


can I buy for, kite —-¢énts pay 
What change, that pencil — €6st 


for one pencil? 
. How many times == canIbuyfor, © 
= poe ae = 
. How many SHILGS is can I buy for, &s): 


Oo ee Ww bw 
= 
<< 
a) 
(om ad 
a 
puad © 
cae 
oo 
= 
) 
oot 
ss, 
eS 
tT 
Gy 
© 
ds 
ot 
~5) 


22), how many can you 


Picture it. 
, how many can you get for, 
adime? What change, over? 


7A wR costs 


How many more cents must he get to buy 


a kite ? 
8. Alice had 65. 


for a eH. How many cents has she now? 


S4 


[el 


* LESSONS. 


ANALYTIC MEASURING EXERCISES, 


@ je| {oe ® © 
: = 'e||@ [@| e = leVta 
Phrases [el : 
to be learned by sight. ie! |e! /e! ° ; 
e| |e 


are equalto, What part, 


es — | 
oN = S 


Oo OTA Dd ow PF WY 


. How many [*!’s are equal to, leel? 
. What part of [eel is lel? 


[eee] is equal to, how many [*i’s? 


. [eee] is equal to, fe *) and , what else? 
. How many [#!’s ,are equal to, [eeee)? 


. How many [*¢!’s are equal to [eeee]? 


What part of leeee|) is lee)? 
[eee e] is equal to eee! and, what else? 
How many [el’s are equal to [eeeee]? 


. (eeeee] is equal to two ¢¢\’s and what? 


(eee! and what else are equal to [eeeee]? 


is equal to [eeee| and what else? 


[e| 


e| (e| 


[e| 
[e| 


. Take away one dot from thislel. What is left? 


like numbers | 


I 


10. 


LESSON XXII. 55 

ee © |e e| i° e © @ @ @ fe 
e je le| e| e| @ e [e| (e| {e| 

e je e @| (fe e je e |e |e {e| 
e e | @| |e e| @| {e| |e je [e 
ee e | e| e je |e (e| (| {|e 
‘e @ je @ [e e ie je |e @| le 
e| je} |e |e |e (ee 


. How many dots must you add to [eeeee| 


. is equal to five and what else ? 


eee and what are equal to Peeee-e]? 


. What two like ‘numbers | are equal to 


[eeeee-e]? What three like numbers ,? 


. What part of is [eee ? Is (ee)? 


eeeee-ee] is equal to @eeee-e and what? 


to make it equal to [eeeee-ee)? 


7. Four and what else are equal to [eeeee-¢e|? 


Two [eee|’s and what are equal to seven? 


eeeee-ee is equal to three [°¢!’s and what? 


What part of [ee eee-ee is fe? 


56 LESSON XXII. 


e ee °) ¢ /e| e| eee ee © © @ 
| e e ’ B e [e| ee e@ (Oe e e | e 
bd ee e e ° [e| e ee e@ @ © © @ 
“ : e| le| s le| e@| e| je; fe] © je fe} fe fe 
le| |e| |e! [el] 2 fe oS ‘ ee fe o oe 
ee e 6 5 e ee e@ @ @ © @ @ 
G e ee e ° /@| e ec ec eo eo wo le ew 
le} |e} [e| je} |e} |e [eo e ee ee e@ © © @| 
Phrases to be learned by sight. [eo] le} le} le! je} je} le} [e 
exactly equal, must you add, 


ee 


. What must youadd, to #eeee-ee) to make 


it equal to [eeeee-eece)? 


bo 


How many ([*!’s are ,exactly equal, to 
jeeceeo-cee|? 


What parti of [eeeee-cee) js [es]? 


oo 


4. What other like numbers, are , exactly equal 
to jeeece-eee|? 
How many more dots in than 


on 


in e@ececee|? 


§. What part of leeeee-eee| js eevee? 


10. 


l; 
12. 


13. 


14. 


15. 
16. 


LESSON XXII (continued). S7 


[eeeee-eee] is equal to two @eel’s and 


what else ? 


. How many more dots in e@eeee-eee) than 


[Se.ceeLe 9 ¢)? 
How many dots must you add to (eeeee-ee 
to make it equal to [eeeee-ceee)? 
[epeeee-ceeee! is equal to six and what? 
[eopee-eeee| js how many more dots than 


p@e@eeee|? 


Two (eeee)s and what else are equal to 
fe :sisisiene 0.00]? 


Find all the like numbers, that are exactly 


equal, to (peeee-ceee|? 
(eee) is what part of leeeee-cecel? 
How many [¢*!’s and what else do you find 


in leeeceo-eee0|? 


58 LESSON XXIII. 


TE AEEEEE 
Tease 
Hoeesisps 


1], [pe eee-ceeee) is equal to nine and what? 


2. What must you add to [eeeee-eee! fo make 


it equal to ten? 
3. Ten is how many more than [eeeee-ee|? 


4, jeeeee-e| and what are equal to ten? 


0. How many [eeeee)’s are equal to ten? How 
. many [¢¢!’s? How many [’s? 

6. What part of ten is [ee|? Is leeeee|? Is jel? 

7. Two (eeee)’s and what are equal to ten? 


8. Three [¢¢e!’s and what are equal to ten? 


LESSON XXIV. 59 


Phrases to be learned by sight and words by sound, 


In change spend dimes 
‘must you get, spent —_ piéc’es 


y) pieces can I get in change, 


pieces can I get |in change, 


? What else? 


Sats 
VS * ON 
a 


get in change, 


He spent 


a er . How many cents in all did 


he spend? How many cents has he left ? 


FP ‘a a: | Es Oe 2 
D. a can you get, for &29: 
Garg tat) re eine a ge = 


60 LESSON XXV. 


EXERCISES IN EQUAL AND UNEQUAL NUMBERS. 
Phrases to be learned by sight and words by sound. 


,equal groups, each group, un’der 


all the sets | write about, be-side’ 


1. Make ten dots in five , equal groups). 


: / Two 
2. Beside , each group , write the name A ‘ 


of the number of dots in it. H two 
3. Under the last group write the RE 
wo 
name of the number of dots in 
H two 


all the groups ). 


4. In the same way make and [3] eve 


,write about, two more sets of | 62 dots. 


equal numbers in ten. 
9. In the same way make and write about 


all the sets, of ,equal groups, in nine 


dots. In eight dots. In six dots. In four 
dots. In seven dots. In five dots. In 


three dots. In two dots. 


LESSON XXVI. 61 


Phrases to be learned by sight. 


unequal groups, —_, both groups, name 


the other sets , the same way, number 


jean 


. Make ten dots in two , unequal groups. 


2. Beside each group, write the name — 
: oie T} 
of the number of dots in it. es 


ww 


. Under the last group write the | 
name of the number of dots in Seat 


both groups | . 


4. In the same way make and , write | t2 dots. 


about, all ,the other sets, of two , unequal 


groups ; you find in ten dots. 
D. In the same way make and write about 


all the sets, of two , unequal groups, you 


can find in nine dots. In eight dots. In 
seven dots. In six dots. In five dots. In 


four dots. In three dots. 


62 LESSON XXVII. 


Words to be learned by sound, 
wv , - bal A = 
orange eut share 


mél’on in’to sup-pose 


1. Into how many parts is this orange cut ? 


ers 


2. What is the name of one part? 


. 3. If I give you one half, how 


melon. Into how many parts 


must I cut it so that they will 


have equal shares? | 


5. This is Ned’s share 


What part 
of the melon is it? What part will be 


Nita’s share? Carl’s share? May’s? 
6. If Carl gives his share to May, how much will 
she have then? Suppose Ned gives May 


his share, too. How much will she have? 


LESSON XXVIII. 63 
Phrases to be /earned by sight and words by sound, 
equal parts break = eake 


my part | each part, eandy eane 


1. Break this candy cane into three equal 
parts. What is the name of each part? 

2. If you give me one third of it and Will one 
third, how much will you have left ? 

3. If I give my part to Will, how much will 


he have then ? 


4. Cut this cake into two equal parts. Name 


the parts. sie Sy 
ok 
5. Cut each half into three VE eg, 2A 
\ SRA 


equal parts. How many 
equal parts in all are there ? 

6. What is the name of each of these parts ? 

7. To how many boys can you give one sixth 


each and have one sixth left ? 


64 LESSON XXIX. 


Words to be learned by sound. 


flock fish find e-rase’ grapes 


May found one nut and Carl found seven. 


What poaae of ees ie May find? Carl? 


4. Here are 2? If you catch one 
fish, what part of all of them will you 
have? What part will be left ? 

9. Make ten dots. Erase one dot. What part 
of all did you erase? What part is left ? 

6. Make ten dots in twos. How many twos are 


there? Hrase one two. What part did 


you erase? What part is left ? 


LESSON XXX. 65 


Phrases to be learned by sight and words by sound. 


Which of us, pay flag  éach 
the most money | paid thus  pié¢’es 


2 
7 BAZ. He paid @ 


2. 
for each dine! How much did he pay for 
all of them? Picture it—thus, J.) 4) 
3. Y) costs two é 


three ‘sleds cost? Picture it. 


4. If I pay three dimes for a 
many can I get for nine dimes? Picture 


eee TAY a AVL Vale 


costs five dimes, how many can I 


get for ten dimes ? 


66 LESSON . XXX. 


Phrases to be /earned by sight and words hy sound. 


at each end how long, inch 
in the middle , still longer | line 
I. ,Make a line, four times ——seizch ___ long. 


How long, is it? Rub out half of it. 


How long, is it now ? 


2. Make a line, six inches long. Put a dot 
at each end, Puta dot in the middle. 


How many times | three inches long | 1s it? 


3. Make a line, two inches long. Put a dot at 


each end. Make this same line two 
inches longer. Put a dot at the end. 
Now make it two inches still longer, and 
put a dot at the end. How many inches 
long is your line now? How many times 
.two inches long, is it? How long is one 


third of it ? 


LESSON XXXII. 67 


* DICTATION EXERCISES ON PARTS. 


Phrases to be learned by sight and words by sound, 


Been Sul0k DEOB Ie el eD 
_the other half. splint paper 


1. Break a match-stick into two equal parts. 
Name the parts. 


2. Give away ,one half. What have you left? 


3. Give away the other half. What have you 
left now ? 

4. How many times did you give away one half? 

5. Cut a slip of paper into three equal parts. 
Name the parts. 


6. Give one third, to Jennie. How many thirds 


have you left ? 


7. Give another third to Jennie. How many 


times have you given away ,one third \? 


How many more thirds have you to give 


away ? 


68 LESSON XXXIII. 


Phrases to be learned by sight and words by sound, 


Can you tell : way apple 


Who can tell, why  éas-i-est 


1. Cut an apple into two equal parts. Name 
the parts. Cut each half into two equal 
parts. How many parts have you now ? 


2. Can you tell, why they are called fourths ? 


3. ‘To how many boys can you give one fourth 
each ? 

4. To how many can you give two fourths each ? 

0. Give away three fourths of your apple. 
What have you left? 

6. Cut a slip of paper into eight equal parts. 


Who can tell, the easiest way to do it? 


7. Group your eighths by twos. How many 
times two eighths have you? To how 


many boys can you give two eighths each? 


10. 


Lb. 


12. 


13. 


14, 


15. 


LESSON XXXIII (continued). 69 


Group your eighths by fours. How many 
four eighths have you? 

If you give me half of all the parts, what 
will you have left ? 

If you give me five eighths, what will you 
have left? If you give me seven eighths? 
Two eighths? Three eighths? 

Break a match-stick into three equal. parts, 


and break each of these parts ,in half, 


How many parts have you? Name them. 


Can you tell, why they are called sixths ? 


If you give two sixths to me and two sixths 
to Carl, what will you have left ? 

If you give away half of all the parts, what 
will you have left ? 

If I give you five sixths of an orange and 
Will gives you one sixth of an orange, what 


will you have? 


70 *LESSON XXXIV. 


Phrases to be learned by sight and words by sound, 


Draw a line, édg’es an’gles 

Are they equal, fac’es edr’ners 

1. How many angles has this fX ? How 
many sides has it ? triangle 


2. Make two triangles. How many angles have 
two triangles? How many sides ? 

3. Make three triangles. What can you tell 
about them ? 

4. How many sides has this Ei How many 

corners has it ? Sea 

9. Make two squares. How many sides have 
two squares? How many corners ? 

6. How many sides has this ? How 
many corners has it ? pént'a-gon 


7. Make two pentagons. Tell all you can 


about them. 


10. 


11. 


12. 


13. 


LESSON XXXIV (continued). 71 


How many faces has this 


ube 


many edges has it? 
Make two etbes and count the faces. The 
edges. The corners. 
Make a square like this ia Draw a 
line from @ to ©. How many triangles 


have you? | Are they equal ? 


Draw a line, from © to @. How many tri- 


angles have you now? (Are these equal, ? 


What part of the square is each triangle ? 
What else can you tell about them? _ 
Make another square like this 


c 


ad 
Draw a line , from @ to » and then one from 


@to@. How many triangles have you here? 


Are they equal? Draw a line from @ to @. 


How many triangles have you now? Are 
these equal? What part of the square is 


each triangle? Tell all you can about them. 


12 LESSON XXXV. 


TABULATED REV/EW 
OF ALL COMBINATIONS IN EACH NUMBER FROM ONE TO TEN. 


Each number being designated in turn, the pupils are directed to 
find and tell; 


/, All the possible additions which make the number, 

2, All the subtractions which can be made from the number. 

3, All the divisions of the number which can be made, both without 
and with remainders. 

4, All the multiplications which form the number, 


ORIGINAL ORAL PROBLEMS. 


Each pupil may be permitted to choose a section, and to tell all the 
combinations he can find within its limits. 


PAIL Ik 


Combinations in Numbers from Ten to Twenty. 


LESSON I. 


LEARNING THE FIGURES FOR NUMBERS FROM ONE TO TEN. 
Words to be /earned by sound. 


groups numbers page show e0py 
a % o three ur - 
* 
seven eight nine ten 
7 8 9 10 


2. What does the word under each group of 
dots tell? The figure under each group 
tells the same thing. 

3. Copy the groups of marks below, and under 
each group write first the word and then 
the figure which tells “how many ” marks: 


e 
T UTND. TTT, TETTTEETD TTETE TD 
TUUUY, VUTTUTT, UUETTTUTT- 
q 


14 LESSON I (continued). 


4. Make the groups of dots these figures call 
fOr VO ed ce ZA ees 

. Put ten sticks or other counters on your 
desk. Tie them into aten. Can you tell 
by figures how many you have? Will it 
do to write only the figure 1 to show that 
you have one ten? 

6. How many counters does | stand for? How, 
then, can you make the figure | stand for 
one ten? | 

The { must be put in the ten’s place, that Js, in the 
second place to the left, to show that it stands 
for one bundle of ten counters, and not for one 
single counter, 

7. Which of these figures, 10, tells the number 

of tens? Which figure tells that there are 
no ones? 

8. Make the figures which stand for eight ones ; 
eight tens; six ones; six tens; five ones; 
five tens; nine ones; nine tens; three 
ones; three tens. 

9. Write the figures for six, four, nine, eight, 
three, seven, two, five, one, ten. 

10. Copy the groups of dots on page 48, and 
under the names of the numbers write the 
figures which tell the numbers. 


Cox 


LESSON II. 75 


LEARNING THE FIGURES FOR NUMBERS FROM ONE TEN TO 
TWO TENS. 


Phrases to be learned by sight and words by sound, 


léarn —ex-det/ly | Work out | 
al’so dif’fer-ence , stands for, 


1. You have now learned the figures that stand 
for all numbers of things from one thing to 
ten things. 

2. You have also learned that we count the tens 
just as we did the ones—thus: one ten (ten), 
two tens (twenty), three tens (thirty), etc. 

3. And also that we use exactly the same fig- 
ures for tens as for ones, the only difference 
being that, when a figure stands alone, it 
means that number of ones, and that, when 
it stands in the second place to the left, it 
means that number of Zens. 

4. We have now to learn to count and make the 
figures by “tens and ones.” If i | i] 
we have this many counters, we 1 
say we have sixteen, meaning six single 
counters and one ten—six and ten, sixteen. 

5. If we have one ten and two single counters, 
we say we have twelve, a word which 
means “two and ten.” 


76 


10. 


diy 


12. 


LESSON II (continued). 


Thirteen means three and ten; fourteen, four 
and ten; fifteen, five and ten; seventeen, 
seven and ten; eighteen, eight and ten; 
nineteen, nine and ten. 

Copy the groups of dots on page 48, and 
under the names of the numbers write the 
figures which tell how many tens and how 
many ones in each group. 


. Put exactly twenty counters on your desk. 


Now count out ten and tie in a bundle. 


What do we call this bundle? (A ten.) 


. Put two single counters to the right of this 


ten. How many in all? Tell by figures 
how many tens and how many ones. 

Work out with your counters and tell first 
by words and then by figures how many 
are 1 ten and 5 ones; | ten and 7 ones; 1 
ten and 4 ones; | ten and 8 ones; 1 ten and 
3 ones; | ten and 6 ones; | ten and 9 ones. 

Tie the single counters you have been using 
into a ten. How many tens have you now? 
Have you any single counters left over ? 

What is the name of two tens? Write 
twenty in figures. What figure do you use 
to show the number of tens? What figure 
do you use to show that there are no ones? 


LESSON II (continued). 7 


13. What number of counters is 1 more than 10? 
What number is 1 more than 13? Than 
lore than 17? ,Than 14? Than 12? 

14. What number of counters is 1 less than 20? 
What number is | less than 16? Than 18? 

15. Group your counters as these figures tell 
you, and write the names of the numbers. 
15, 16; LS LAlog dh Tabl ots, 20519: 


APPLICAT/ONS. 


1. Which would you rather have, 8 pennies or 
80 pennies? 9 dollars or 30 dollars? One 
dime or nine cents? Why? 

2. In 19, which figure stands for the greater 
number of things, the 1 or the 9? Why? 

3. Would you rather have the number of tardy 
marks the 1 stands for, or the number the 
9 stands for? Why? 

4. Nat and Will have 17 nuts. Nat has as many 
as the 1 stands for, and Will as many as the 
7 stands for. Which has the greater num- 
ber of nuts ? 

5. In 11 the figures are alike. Do they each 
stand for the same number of things? 
Why not? What is the difference? 


— 


78 LESSON III. | 


LEARNING THE SIGNS + AND =. 
Words to be learned by sound. 


shorter sion fig’ures 
writing word — sén’ten-ces 


1. How many are and Bae 


Four and five are nine. 
2. Here is a shorter way of writing this: 
4+5=9, 
3. What word is the figure 4 used for? 
What word is the sign + used for? 
What word is the sign = used for? 
4. Write these sentences and put figures and 
signs in place of the words and the dashes: 
Three and five are ——. 
Seven and two are ——. 
Nine and one and five are ——. 
Seven and three and six are ——. 
Four and six and eight are ——. 
Three and two and seven and one are ——. 
Six and four and two and three are ——. 
Five and six and seven and one are ——. 
Kight and two and two and two are ——. 
Ten and three and three and three are ——. 
Six and three and one and eight are ——. 


*LESSON IV. 79 


ADDITIONS BETWEEN 1/0 AND 20, 
Phrase to be learned by sight and words by sound. 


,all the additions | sin’ole — signg 

1. Put 2 single counters beside a ten, thus: 
In which place have you put the ten ? I | 
In which the single counters ? 

. What number is this? Tell by figures. 

3. Add three counters to your 12 counters, then 
tell by figures and signs how many are 12 
counters and 3 counters. Let the letter c. 
stand for counters. 

Thus: 12¢.+3¢.=15c. 

4. Work out with your counters and tell in the 
same way how many are 12c. and 4c¢.; 
12c. and 6c.; 12c. and 8c¢.; 12c. and 5c.; 
12¢c. and 7c.; 12c. and 1c. 

Dd. Make a 13 with your counters. Add 7 
counters. Tell by figures and signs, thus: 
13+7=20. Tell in the same way how 
many are 13 and four; 13 and six; thir- 
teen and 5; thirteen and three. 

6. Make a 14 with your counters, and work out 
with the counters you have left over and 
tell by figures and signs all the additions 
you can make. 


bo 


80 LESSON IV (continued). 


7. Work out with your counters and tell by 
figures and signs how many you must add 
to 15 to have 18; how many you add to 
15 to have 20; to have 17; to have 19. 

8. In the same way find and tell how many you 
must add to 16 to have 20; to have 18; 
to have 19; to have 16. How many must 
you add to 17 to have 20? How many 
must you add to 18 to have 20? 


ee ag eee ee 
LESSON V. 


CONCRETE APPLICATIONS. 


1. Jane has a dozen eggs. How many more 
must she buy to have 16? 

2. I rode 14 miles in the train and 4 miles in 

the stage-coach. How far did I go? 

. May read 15 pages in her reader last week, 
and she has read 4 to-day. How many 
pages in all has she read ? 

4. I spent 13 cents for a kite, and 4 cents for a 
string, and had one cent left. How many 
had I at first ? 

5 Carl has 16 cents in his bank and 3 cents in 
his pocket. How much money has he? 


ww 


16. 


17. 
18. 


LESSON V (continued). 81 


. There are 11 goats on the hill-side and 5 goats 


by the road-side. How many in all? 


. How many are 13 lilies and 3 lilies? 
. There are 12 nuts on the tree and 5 nuts on 


the ground. How many altogether ? 


. On one branch I can count 14 bees and on 


another 5 bees. How many on both? 


. How much money is 11 dollars and 7 


dollars ? 


. Carl has 15 doves and May has 3. How 


many doves have they both? 


. | have 14 buttons, but I need to have 17. 


How many more must I get? 


. I can see 12 ships sailing down the bay and 


7 ships sailing up the aS How many in 
all do I see? 


. How many are 15 flags and 4 flags? 
. In the pond there are 11 frogs and on the 


bank there are 3. How many in all? 

[ have made 17 marks on my slate and have 
3 more to make. How many shall [ have 
made then? 

How many are 15 cherries and 6 cherries ? 

I have a dozen marbles, Nat has half a 
dozen, and Will has 2. How many have 
we altogether ? 


82 LESSON VI. 


OBJECT AND SLATE WORK. 
Work out with counters, copy and complete. 


12+3= lo 2 = 14+3= 17+2= 
13+2= 12+5= 137+4= 12+7= 
14+5= 13°F 6> 16+1= 144+2= 
1+4= 16-3'= l1l+6= 25 4 


Copy, complete, and afterward prove with counters. Try to do them 
in five minutes. 


1+-4= 4-5 = ede d+9= 
11+4= 14+9= 1h Ge ee i ioe ts) = 
Ap age bis o+4= 7+1= oS 


14+1= 1l+4= IB fos pet ce: 1+3= 


* ORIGINAL PROBLEMS, | 
Write or tell number stories about anything you like, using: 


3+3 12+5 13 +6 ie ayellak yy 
ei) LO 12+5 14+3 beyond 


RAPID ORAL CALCULATIONS, 
Read off quickly, giving the answers at once, 


aL 2 ela 13+3 1444+4 15+5 1Atte Td 


8+16 44+11 5+12 174+3 1941 
10+4 5+13 16+4 18+2 1347 
5+14 15+4 8+11 134+6 1742 


LESSON VII. 83 


LEARNING THE SIGN —. 
Words to be /earned by sound, 


more than léss ex-préss’ dash’es 


ew) 


~] 


. How many are 14 less 3? 16-—4=? 17-5= 
. How many more are 13 than 10? 15—-3= 


e@ 98 e@ ..°¢ 
. How many are less three dots ? 


Eight less three are five. 


. 8—3=5 is a shorter way of writing this. 
. What word is the sign — used for? 
. May has this many pencils, |[|]], and Carl 


has this many, |]. How many more pen- 
cils has May than Carl? 
Five 1s three more than two. 


. 5—2=8 is the way we express this in figures 


and signs. 


. Write these sentences with figures and signs 


in place of the words and dashes: 


Seven less five are ——. Ten is —— more 
than six. Sixteen less six are ——. Nine 
is —— more than seven. Twelve less two 
are ——. Seventeen is —— more than 
seven. Eighteen less nine are ——. Nine- 
teen is —— more than one. 


9 
Pha 
Toso? 190 =P 16-3 =? 


84 


*LESSON VIII. 


SUBTRACTIONS BETWEEN 0 AND 20. 
Phrase to be /earned by sight and words by sound, 


,all the subtractions , un-tying _fig’ures 


ie 


2. 


with-out’. eount’ers 


Put one bundle of ten and ten single counters 
on your desk. How many in all are there? 

From your 20 counters take 8 counters, and 
tell by figures and signs how many are left. 
Thus: 20 c.-- 8. = 12ie. 


. In the same way find and tell how many are 


20 c. less 6c.; 20c. less 3c¢.; 20c. less 5c. ; 
20 c. less 9¢.; 20c¢. less 7c.; 20c. less 2c. ; 
20 c. less 4¢.; 20c. less Le. 


. Have 18 counters on your desk. Now find 


and tell how many are 18 counters less 8 
counters; 18 less 6; 18 less 4; 18 less 2; 
18 less 7; 18 less 5; 18 less 3. 


. Have 17 counters on your desk. Find and 


tell all the subtractions you can make 
without untying the ten bundle. 


. Have only 16 counters on your desk. In the 


same way as before, find and tell all the 
subtractions you can make. Do the same 
thing with 15 counters; with 14; with 13; 
with 12; with 11. 


LESSON IX. 85 


CONCRETE APPLICATIONS, 
Phrases to be /earned by sight and words by sound, 


, this morning | ate broke sehool 
, this afternoon , éat brok’en _ stable 


1. May had two dimes. She spent one cent for 
a paper doll. How much money had she 
left ? 

2. Carl, too, had two dimes, and he spent 5 
cents fora top. How much had he left? 

3. After Carl had bought his top he lost 4 of his 
cents, and then he put what he had left 
into his bank. How much did he put into 
his bank ? 

4. May gave the rest of her money, all but 6 
cents, to Will and Nita. How much did 
she give away? 

5. Nell had a dozen and a half of eggs in a 
basket. She let it fall and broke 7 of the 
egos. How many were not broken? 

6. 17 goats less 6 goats are how many ? 

7. On his way to school Tom found 14 nuts. He 
ate all but three. How many did he eat? 

8. This morning there were 19 cows in the 
stable; now there are only 15. How many 
have been taken out? 


86 


oO 


10. 


11. 


14, 


15. 


16. 


LESSON IX (continued). 


. This afternoon Jack had 16 bags of pop-corn 


to sell. He sold only five. How many 
has he left? 

Jennie’s little white hen sat upon 15 eggs. 
All hatched out but 2. How. many little 
chickens has she ? | 

This afternoon Will and Walter found 19 
pond-lilies. Will found 8 of them. How 
many did Walter find ? 


. There are 18 trees in a field. 5 are apple- 


trees and the rest are plum-trees. How 
many plum-trees are there? 


. John had 17 marbles and lost 4. How many 


has he left? 

Nat is 13 years old and Will is 11. How 
much older is Nat than Will? 

May has 16 buttons and Nita has 13. How 
many more has May than Nita? Picture 
it, thus: 

May’s buttons SPETEETTEE oe. ee0 


Nita’s buttons [ee ee0e-ceeeeolece 
/6 is 8 more than (8. 


Carl had 19 chestnuts. He gave his squirrel 
7 of them. How many has he left?  Pict- 
ure it, thus: [ee eee-coccelececccces 

19 less 7 are (2, 


LESSON X. 87 


OBJECT AND SLATE WORK. 
Work out with your counters, copy and complete, 
20 -—-7= 19-—9= L3 — lis l1l-l= 
Lin3 = 14—2= 12-2= 18—-4= 
Picture these with dots (as in example /5, page 86) and copy, giv- 
; ing the answers, 

How many more are 

io than*s ? 15° than’3 ? 18 than 6? 

14 than 1? 16 than 2? 17 than 3? 


* Copy, complete, and prove with counters, 


i—?=6 20-—?=7 14-3=? 15—?=12 
17—6=? ?-17=2 ?—-T=11 13-3=? 


RAPID ORAL CALCULATIONS. 
Read and give the answers at once. 


10 less4=? 9less3=? 6 less 4=? 13 less 2=? 
ZO AK? 19 Hs B=? 16)-— 457: 18':- 8=? 
eet a 1 20) = 8S Pld = oS? 


What is the difference between 
19 and 13? 15 and 12? 14 and 20? 
14 and 17? 2 and 10? 16 and 1? 
How many more are 
Piethan elo? 16ithan 313.2, 19;than 12? 
12 than 10? 18 than 5? .20 than 17? 


88 


LESSON XI. 


MAKING UP TENS. 


. Group your counters like these marks: |]]][]], 


Hi]|. How many counters in the first 
eroup? How many in the second group ? 


. Make a ten of the first group by adding 


counters from the second group; thus: 
WME 

ow many counters did you put to the seven 
to make ten? Then how many of the 9 
are left? 


. Show BOY figures what you have done; thus: 


(+342=12. Then i-9 = 


. Place your counters in groups of 6 and 9; 


thus: |FULL, TIPLEE, |. Make a ten of the 


six with counters taken from the nine; 


thus: TIETE TTD TE 


. How many counters did you add to the 6 to 


make a 10? How many of the 9 are left? 
Then 6+44+5=? 64+9=? 


. Place your counters in groups as these figures 


tell you, then add, making a ten of each 
first group. Write every example both 


ways; thus: 7+34+3=18. TOS 133 


Tet O44S-+D! 9b 4S 9) 54 Oa Ge ae 
CTT UA Be 6 20 8 Be 9 SEO BESET ay 


LESSON XII. 89 


CONCRETE APPLICATIONS. 
Words to be learned by sound. 


both pair oar’den write used 
a-20' shoes rose’bush wrote doz’en 


1. In my garden there are 6 roses on one rose- 
bush and 5 roses on another. How many 
roses are there on both bushes? 

2. Carl wrote 8 words on his slate this morning 
and 4 this afternoon. How many in all 
did he write? 

3. If there are 7 buttons on one of little Nell’s 

shoes, how many are on the pair? 

Nine roses and 5 roses are how many? 

In one box there are 9 pencils and in the 
other box there are only 2. How many 
pencils in both boxes? 

6. Jane used a half a dozen eggs this morning 
and a half a dozen this afternoon. How 
many in all has she used to-day ? 

7. There are 8 bees on the bee-hive and 7 bees 
on the stand. How many altogether ? 

8. Four years ago Ned was 9 years old. How 
old is he now? 

9. If | buy 7 marbles and you buy 5, how many 
shall we both have? 


ie 


LESSON XIII. 


90 
Words to be learned by sound, 
spider fiéld an-oth’er al-to-géth’er 
1. Will gave me 8 daisies this morning and 5 


10. 
1a: 


this afternoon. How many daisies has he 
given me to-day? 


. How many are 9 cents and 6 cents? 
. Carl has drawn a line 7 inches long. How 


many inches must he add to make it 11 
inches long ? 


. Carl has drawn another line 9 inches long. 


How many inches must he add to make it 
a foot long? 


. How many legs has a fly? How many legs 


has a spider? How many legs have a 
spider and a fly together ? 


. How many are 7 frogs and 6 frogs? 
. There are 9 lambs in one field and 8 lambs 


in another. How many in both? 


. If you have three cents and I give you 8 


more, how many will you have? 


. Nine blocks and how many more are 16 


blocks ? . 
Hight dots and how many more are 16 dots. 
If there are 9 buttons on one of your shoes, 
how many buttons are there on the pair? 


LESSON XIV. 91 


OBJECT AND SLATE WORK. 
Work out with counters, copy and complete. 


7+5=? 8+5=? 9+9=? 
?+6=11 9+ ?=11 r+7=14 
8+ ?=14 ?+5= 14 8+8=? 
9+7=? 8+ ?=15 9+ ?=16 
Copy, complete, and prove with counters. 
6+6=? °+7=15 7+?=18 
7+9=11 9+4+8=? 8+4=? 
4+9=? 6+ ?=18 ?+6=15 
?+6=15 8+3=? 6+5=? 


ORIGINAL PROBLEMS. 
Make as many problems as you can of these, 


249=16 ?+?=18 2+?=13 
40 =17 ?+?=15 P+?=12 
o4+9=14 2+?=19 2+? =20 


RAPID ORAL CALCULATIONS, 
Read and give the answers at once. 


10+10= 84+5= 4+9= 
8+ 8 = 6+9= 7+9= 
T+7 = 7+6= 8+6= 


I+ 9 = 8+7= 9+95= 


92 


1. Place 13 counters on your desk, the 


LESSON XV. 


BREAKING UP TENS. 


10 tied in a bundle and the 3 single {J | 
sticks beside it. Take 9 counters from 
these. Can you take 9 counters from the 
3 counters? How many of the 9 can 
you take? What must Bek cee do to get the 


rest of the 9? Mien Ime 3—6=—7 [ose 


— 
. Place your counters as these figures tell 


you (the ten tied always in a_ bundle), 
From each number take 6. Write your 


work both ways; thus: 15-5—-1=9. 
15 — 6 Dagel2ad Its: 


. Place your counters as these figures tell you, 


and from each number take 7. Write 
your work both ways. 16, 13, 12, 15, 11, 
Zs 


. From each of these numbers take 9. Write 


your work both ways. 17, 138, 16, 11, 16, 
12, 15, 18, 14. 


. From each of these numbers take 8. Write 


your work both ways. 11, 12, 18, 14, 15, 
Noy y IEG. 


. From each of these numbers take 5. Write 


your work both ways. 15, 14, 13, 12, 11. 


LESSON XVI. 93 


CONCRETE APPLICATIONS. 
Words to be learned by sound, 


léarn béans  péas’ length  dif’fer-ence 


Hh 


2. 


May had 13 buttons on a string. She lost off 
7. How many has she left ? 

Fred is 16 years old? How old was he 8 
years ago? 


. My slate is 11 inches long and May’s is only 8 


inches long. What is the difference in 
length between them ? 


. Carl drew 17 birds on his slate, but erased 9 


of them. How many has he left? Picture 
it, thus: Peeeeesedasossees 


{7 lessG4 aré 22e., 


. Nita has a dozen blocks and Will has half a 


dozen. How many more blocks has Nita 
than Will? Picture it, thus: 


Nita’s blocks [ee eeeeeee cleo 
Will’s blocks eccooeocee 


[2 are =2.a\more than 6: 


. Fred planted 11 beans in his garden, and 


9 peas. How many more beans than peas 
did he plant? Picture it. 


. There are 15 birds in a tree by my window. 


If 8 fly away, how many will be left ? 


94 


LESSON XVI (continued). 


. In the pencil-box there are 14 sharp pencils 


and 8 dull ones. How many more sharp 
pencils than dull ones? 


. I see 11 frogs on a log. If 6 hop away, how 


many will be left ? 


. Carl had 13 words to learn. He has learned 


8, how many has he yet to learn? 


. How many more are 15 rabbits than 9 rab- 


bits ? 


. Tam 12 years old and Carl is 7. How much 


older am I than Carl? 


. May made 16 paper-dolls and gave 9 of them 


to Nita. How many did she keep? 


. Carl had 14 doves. 7 of them flew away. 


How many has he left ? 


. Four months of this year are past. How 


many are still to come? 


. There were 11 ducks in the pond. 7 have 


come out. How many are still in the pond ? 


. Fred made 14 kites. He sold 9 of them. 


How many has he left ? 


. If Jane buys a dozen oranges and gives 


away 8, how many will she have left? 


. John had a dozen and a half pop-corn-balls. 


He sold 9. How many has he yet to sell? 


. How many are 13 apples less 9 apples ? 


LESSON XVII. 95 


OBJECT AND SLATE WORK, 
Work out with counters, copy and complete, 
1l—6=? ?—6=6 I17-9=? ?—-8=8 
13-—?=6 16-9=? ?-9=5 12-9=? 


Picture with dots, as in Lesson XVI, 


How many more are 
11 than 7? 16 than 7? 15, than 6? 
17 than 8? Teese 12 than 5? 


Copy, complete, and afterward prove with counters, 


15— ?=6 12—-—9=? °—9=9 
11-—8=? ?—d9=18 dE Deere Breese 
a 16—/%=—8 fo (at 


RAPID ORAL CALCULATIONS, 
Read and give the answers at once. 

How many are 

10 less 9 ? LOai6 2 lou ier iL sehes 

Crees SS VEPs Ding Leo? 13s 97 
What is the difference between 

9 and 13? 3 and 12? 14 and 6? 

16 and 17? 17 and 8? 18 and 9? 
How many more are | 

15 than 8? 18 than 9? 16 than 9? 

17 than 9? 13 than 6? 17 than 8? 


96 LESSON XVIII. 


MULTIPLICATION. 
LEARNING THE SIGN X, 
Words to be learned by sight and sound, 


times place an-oth’er showing 


1. May wrote two words this morning and two 
words this afternoon. How many times did 
she write two words? How many in all? 

Two times two are four. 

2, 2x 2=4 1s the shorter way of writing this. 

. What word is this sign X used for? 

4. How many mittens are there in a pair? How 

many in four pairs? 
Four twos are eight. 

. Four 2’s = 8 is the shorter way of writing this. 

6. Write these sentences with figures and signs 
in place of the words and dashes: 


Co 


Or 


Three times three are ——. Six twos are ——. 
Four times five are ——. Three fours are ——. 
Two times four are ——. Five threes are ——. 
Nine times two are ——. Two eights are —_. 
Six times three are ——. Four fives are ——. 


7. How many are 2X5? 6X2=? 10X1=? 
2X8=? 9X2=? 6X3=? 

8. How many are two fives? Three fours =? 
Two 8's =? Nine l’s =? Seven 2’s=? 


dees 


al, 
L2. 


~) 


LESSON XIX. 97 


EXERCISES IN MULTIPLYING. 


Group 20 counters in 2’s, thus: |] |] qT] UT] UI 
HUE UT Up dy.) Plow many 2’s are there? 
Find with your counters and tell by figures 
and signs how many are ten 2’s. Three 
2’3. Five 2’s. Seven 2’s. Nine 2’s. Two 

2’s. Four 2’s. Six 2’s. Hight 2’s. 

Group your 20 counters by 4’s, thus: |] ]] {II 
LEE VUTL UIE. How many 4’s are there? 
How many areo x4? 2x4? 4x4? 3X4? 
Group your counters in 5’s, thus: ||]]] Tf] 

PUTT UEP]. How many 5’s are there ? 

How many are four 5’s? Two's? Three 5’s? 

Group your 20 counters in 3’s, thus: |]] [jf] 
HE LVL VTE ULE df.) low many 3’s are 
there and what else ? 

Put aside the two counters left over. How 
many counters have you now? 6X3=? 
How many are 4X3? 3X3? 9X3? 2X8? 
Group your 20 counters by 6’s. How many of 
them do you need to make even 6's? Put 
the others aside. How many 6’s are there? 

How many are two 6’s? Three 6’s? 

Group your counters by 9’s. How many of 
them do you need to make even 9’s? How 
many are two 9's? 

0 


98 


LESSON XX. 


CONCRETE APPLICATIONS. 
Words to be /earned by sound, 


prompt éarn old least 
réal-ly léarn old’er most 
bréak’fast vérse dld’est month 
1. Every day for one week May was given 2 


cents for being prompt at breakfast. How 
many times two cents had she at the end 
of the week. How many cents in all? 


. Carl was given 3 cents for every basketful of 


weeds he picked out of the garden. How 
much did he get for 5 basketfuls ? 


. When eges are 3 cents each how much will 


a half dozen cost ? 

May was to get 4 cents for every verse she 
learned. She learned one verse every week 
for a month. How much did she earn? 

Carl, who is older than May, was to get 2 
cents for every verse he learned. He 
learned 2 verses every week for a month. 
How much money did Carl earn? 

Nell, who is the oldest, was to get only 1 cent 
for each verse she learned. She learned 
3 verses each week for a month. How 
much did Nell earn ? 


10. 


a), 


12. 


15. 


14. 


15. 
16. 


LESSON XX (continued). 99 


Which of these children earned the least 
money? Why? Which one learned the 
most verses ? 


. Jack made 2 large kites. Fred gave him 2 


new lead-pencils for each kite. How many 
pencils did Jack get ? 


. Fred had just paid 5 cents each for the lead- 


pencils. How much money did the kites 
really cost him ? 

Nellie bought 3 apples at 2 cents each, and 
2 lemons at 4 cents each. How much did 
she pay for the apples? How much for 
the lemons? How much for both lemons 
and apples ? 

Nita wrote 2 rows of words on her slate, and 
there were 5 words in each row. How 
many words in all did she write ? 

May wrote 2 rows of words of 10 words each. 
How many words did May write ? 

In my spool-box there are 3 rows of spools 
of 3 spools in each row. How many in all? 

Annie has 2 such boxes of spools. How 

many spools has Annie ? 

How many legs have half-a-dozen sparrows ? 

My desk is 2 times 8 inches wide. How 
wide is it? 


100 LESSON XXI. 


OBJECT AND SLATE WORK. 


Work out with counters, copy and complete, 


2 threes = ? y Aemaesitee ie ean | by 3 fours=? 

? twos = 14 ? threes = 15 ? sixes = 18 
4. PWas 20 5 twos =? 8 ? =16 
Copy, complete, and afterward prove with counters. 
2xX2= 2xX9= 2x 10= 
3X3= aX3= 5X 4= 
TX2= 4x4= 4x 2= 
aX6= 2xX8= 2X 6 = 


ORIGINAL PROBLEMS. 


Make as many problems as you can from these. 


?x?=12 Perea oa LS rx?=16 


RAPID ORAL CALCULATIONS, 


Read and give the answers at once. 


3 twos are 7 twos are 4 fours are 
5 fours are 3 fives are 5 twos are 
2 fives are 6 twos are 2 eights are 
Dex 4 10.42) = 5X Qier 
bo<eo 4.393t= DIK Sas 


4X5= oxXx5o= 6X3= 


LESSON XXII. 101 


DIVISION—CASE /, 


Finding out how many parts there are in a number when we know 


wits 


what one of the parts is, 
LEARNING THE SIGN =~. 


If we have 10 marbles, to how many boys can 
we give 2 marbles each ? 

How many marbles are we to give to each 
boy? Then 2 is the number in each part, 
and what we want to find out is, how many 
of these parts there are. 

Let us find how many 2’s in 10 marbles, thus: 

ee ee ee ee ee 

How many parts do we find there are? Then 
how many boys will get 2 marbles each ? 

/n ten there are five 2’s, 

The shorter way of writing this is: 10+2’s=5., 

For what words is this sign + used ? 

Write these sentences with figures and signs 
in place of the words and dashes: 

There are —— fours in eight. There are —— 
threes in nine. There are —— fives in ten. 
There are —— fours in twelve. There are 
—— threes in fifteen. There are —— fours 
in sixteen. There are —— sixes in eighteen. 
There are —— nines in eighteen. There 
are —— fives in twenty. 


102 LESSON XXIII. 


CONCRETE APPLICATIONS. 
Words to be learned by sound. 


eould séa’son di-vide’ péo’ple 


would va-ea tion ean’dleg ear’-fare 


Te 


2. 


If Jane pays 18 cents for half-a-dozen candles, 
how much is that for each candle ? 

May earned 18 cents making lamp-lighters, at 
3 cents a dozen. How many dozen did she 
make? 7 

If May gets 3 cents for a dozen lamp-lighters, 
how many does she make for one cent ? 

Carl bought 16 marbles for 4 cents. How 
many marbles could he get for 1 cent? 


. If you pay 16 cents for 8 lemons, how much 


would one lemon cost you? 

Our school had 14 days vacation at Christmas. 
How many weeks was that ? 

May had a dime and a five-cent piece. She 
changed them for three-cent pieces. How 
many three-cent pieces did she get? 

How many two-cent stamps can you buy for 
two dimes ? 

IT have a dozen and a half nuts to divide 
among 9 boys. How many can I give to 
each boy? 


10. 


‘1. 


LESSON XXIII (continued). 108 


The 12 months of the year make the four 
seasons. How many months to each sea- 
son? What are the names of the seasons ? 

If a paper doll costs 2 cents, how many can 
you buy for 8 cents? How many for 14 
cents? For 10 cents? For4cents? For 
6”? (¥ stands for cents.) 


. May has a spool-box with nine spools in it. 


There are three spools in each row. How 
many rows are there ? 

I spent 20% for car-fare to-day. How many 
times did I ride? (Fare, 5 cents.) 

How many times could [ ride for 10%? 


. How many times could I ride for 15%? 


How many times could you and I ride to- 
gether for 20%? 


. Where Carl lives, the fare is 6% for grown 


people, children half-fare. How many times 
could Carl and his father ride for 18 ¢ ? 

A yard is equal to 3 feet. Then how many 
yards long is our hall, which is 15 feet long? 

Carl’s fish-line is 12 feet long, how many 
yards is that? 

The door is 8 feet high. How many yards 
high, and how many feet over ? 


. How many yards are equal to 18 feet ? 


_— 


104 LESSON XXIV. 


OBJECT AND SLATE WORK. 


Work out with counters, copy and complete. 
20 + 4’3 =? PIs 10 So 
16+ ? =8 10+ 2’s=? P+ 3’3=—4 
Copy, complete, and afterward prove with counters. 


There are 
Pal sian la PHOS ne U2 2 xe SATB LeS 
107 nv 6 A picvelnet) re Dash el ieer) 


ORIGINAL PROBLEMS, 


Make as many problems as you can from these, 


18+?=? 20+?=? Sar? 
16+?=? Le 14+?=? 


RAPID ORAL CALCULATIONS, 


Read and answer at once, 


How many 
3’8 in 6 ay ann, 4’3 in 8 
D’s in 20 28 in: 4. Postel 
Us in 14 4’3 in 16 Hy Suinelo 
12+ 6's= 20 = 10'3'= 12-4 9= 
14+2’s= 10+ d’s= 15+3’s= 


9 = 3’3 = Si G2 6=+2’3= 


LESSON XXV. 105 


DIVISION—CASE //, 


Finding out what is one of the equal parts of a number when we 


a 


cS Ot 


know how many parts there are, 


LEARNING THE FRACTIONAL EXPRESSION, 


Little Annie has 15 nuts. She says she will 
share them with us—you, herself, and me— 
if we will find out what her share will be. 
There are three of us, so the nuts must be 
divided into three equal parts. 


. Now we know how many parts there are, and 


what we want to find out is what is one of 
the parts. 

There are five 3’s in 15, and if we take 9eeee 
1 from each 3 we shall have !/; eeeee 
of all of them. Then Annie’s share will be 
how many nuts ? 


One third of fifteen is five. 


. V/s of 15 is 5 is a shorter way of writing this: 
. For what words is !/3 used ? 
. Work out these examples with your counters. 


Write them in the shorter way: 
What is one half (1/2) of six ? 
What is one third (1/3) of nine? 
What is one fourth (1/4) of sixteen ? 
What is one sixth (1/s) of eighteen ? 
What is one tenth ('/j0) of twenty ? 


106 LESSON XXXVI. 


UNEQUAL DIVISIONS, 


1. What is one third of ten? I have ten sticks 
of candy to give to three boys. How must 
I divide it so that they will have equal 
shares ? 

2. By giving one stick at a time to each boy, 
I find that when I have given 3 sticks to 
each boy I have | stick left. This may be 


pictured thus: i HI {I . 
B B B 


3. Now, to divide this one stick equally among 
them, I must break it into three equal 
parts—thirds—and give one third to each 
boy. 

4. Ten sticks divided into three equal parts may 
be pictured thus: []], [I], TJ... 19+38= 
31/3, and one of these equal parts, or one 
third, will be 31/3; 1/3 of 10=31/3. Then 
each boy’s share will be 31/3 sticks of candy. 

9. What would be two of the boys’ shares put 
together ? 2/3 of 10 =? 

6. In the same way, work out with counters, and 
express in figures and signs, how many are 
7 + 3, also 1/3 of 7 and 2/; of 7; 13+ 3, also 
1/, of 13; 16 +3, also 1/3 of 16 and 2/3 of 16; 
19+ 3 and !/s of 19. 


LESSON XXVI (continued), riay, 


7. What is one half of nine? Take nine count- 
ers, and find first the largest number in 
nine that can be divided into two equal 
numbers. What have you left over? Now 
divide this one counter equally between the 
two groups. Then nine divided into two 
equal parts may be pictured thus: |]]}, 
Ml. 92 = 4%. 

8. And one of these equal parts, or !/. of 9=? 

9. In the same way, work out with counters, and 
express In figures and signs, how many are 
o+2 and '/, of 5; 11 +2 and 1/, of 11; 
15 + 2 and !/p of 15; 19+ 2 and 1/5 of 19. 

10. What is one fourth of thirteen? Take thir- 
teen counters, and find first the largest 
number in thirteen that can be divided into 
four equal numbers, then divide the one 
counter left over equally among the four 
oeroups. Then thirteen sticks divided into 
four equal parts may be pictured thus: 
HW, WW, WW. 18 +4344. 

11. And one of these parts, or !/, of 13 =? 

12. In the same way, find with counters and ex- 
press in figures and signs how many are 
9+4 and 1/4, of 9; 17+4 and !/ of 17; 
also, 1/,0f 5; '/5 of 11, of 16; 1/6 of 13, of 19. 


108 LESSON XXVII. 


CONCRETE APPLICATIONS. 
Words to be learned by sound. 


orand moth-er black’board pécks 
lamp’light-ers black’bér-riegs quarts 


she 


2. 


3. 


J 


May has 12 cherries, but Carl has only half as 
many. How many cherries has he ? 

Nita has only one third as many cherries as 
May. How many has Nita? 

Will has !/, as many as May. How many 
cherries has Will? 


. Bertha has only '/6 as many cherries as May. 


How many has Bertha? ~ 


. There were 15 blackberries on a bush. Tom 


took '/; and Fred took 1/3 of them. How 
many did each boy take? 


. I had a plate of apples. I gave !/, of them 


to May and !/, to Nita. How many 4ths 
had I left ? 


. [had 8 apples left. How many had I at first? 
. May made 17 lamp-lighters. She used one and 


gave '/> of the rest to Jane. How many 
did she give to Jane? 


. Will wrote 6 words on the blackboard, which 


was !/3 of the lesson. How many words in 
all the lesson ? 


10. 


14, 


15. 


21. 
22. 


LESSON XXVII (continued). 109 

IT had 20%. I spent !/ of it for car-fare, and 
gave '/; of the rest to May. What did she 
get? 


. [had 16 dimes. I gave !/s of them to May. 


How many dimes had I left ? 


. I spent 1/2 of the dimes I had left for a sled. 


How much did my sled cost ? 

Carl has $18. His father gave him !/s of it, 
and his mother gave him !/9 of it. How 
many dollars did each give him? (The 
sion 5 stands for dollars.) 

His grandmother gave him !/, of it, and Carl 
earned the rest. How much did he earn? 

There are 8 quarts in 1 peck of peas. How 
many quarts in a half-peck ? 

How many quarts in a peck and a half? 


. Carl sold 6 quarts of nuts last fall. What 


part of a peck was that ? 

There are 12 inches in a foot. My reader is 
'/, of a foot long. How many inches long? 

My copy-book is */, of a foot wide. How 
many inches wide ? 

Carl’s tool-box is 11/, of a foot long. How 
many inches long is it ? 

What part of a foot are 9 inches? 6 inches? 

My room is 5 yards long. How many feet? 


EO LESSON XXVIII. 


OBJECT AND SLATE WORK, 


Work out with counters, copy and complete. 


1/4, of 20 =? Ot leas V4 of 12 =? 

erOt 2 8G Ue oft) = 3H ? of 16=4 
'/g of a= 2 1/9 of 18 =? M/s of 2? =3 
eotalye= 2 Lobe 1/5 of 16 =? 


ALIS) 1/o Of 2 YO 1/4 184 /g Of 2 O24 6s Aye One 
Copy, complete, and prove with counters, 


1/5 of 11 =. 1/7 of Ideal 4/5 of LOS 1 /sc0t 13 
We of 12 = 41/g of LT = et/eiof 1d maths oF 19 


ORIGINAL PROBLEMS. 


Make as many different problems as you can of these, 


aay e. Wh eset Cit eye > 1-Olsba 
Ov We Sy VO e—er Mn) bite = 


RAPID ORAL CALCULATIONS, 


Read and answer at once. 


Vsof20= 'Yso0f12= Yeof 9= 3/5 of 15= 
Wg of 19= Ysofll= ',0f 1I8= 1/4 of l2= 


oust Y/gio£,.: Bis) vote ids Ota aust. or 
Dist/,of 418 1/3 of 9is 1/5 of 10 is 1/, of 


LESSON XXIX. pis bp | 


REVIEW. 

Analysis and synthesis of each number from /0 to 20, 
Bt 

meee Steed W-38=? | 84+27=11 

?+5=11 64+5=? ?—2=9 11-4=? 

How many more are ll than8? Than2? 5? 7? 

How many less than ll are 4? Are 3? 9? 6? 

How many must you add to 7 to make 11? To 
6? To4? To 2? 

Tie your counters together by 2’s, and see if 
you can make 11 with 2’s. Try the same 
thing with 3’s; with 4’s; with 5’s. 

Take 11 counters and see if you can divide them 
into groups of 2’s; of 3’s; of 7’s; of 8's. 

eos). U2? lle rs otf, Lr 2b 


12 
2+10=? 12-4=? 34+?=12 74+5=? 
Peer = 12-2 —? ey12+ 6 =? 12+ 4% 
Bee teat 2S S19) 4 xs? 12 12-89 
WP H4 WfokF 8 oP 66 6 2x GR? 
Ones? GHt = SP0f12S6 “tyor lar 


What is the difference between 12 and 8? Be- 
tween 12 and 7? 6and12? 9 and 12? 


What must you add to 10 to make 12? To 8? 
Tote 16.8? 


i sep LESSON XXX. 


13 
18-37 OP PES Sot UO caeae Org 
Ooh =o 13 4 ee e— 6. a ee 
Poh) Lo 2 3 a ar eee 
Moe lol 42-7213) 2a OS oe acne 
How many less than 13 are ll? Are 12? 9? 6? 
How many more are 13 than4? Than 8? 12? 7? 
Tie your counters together by 2’s. Try to make 
13 with 2’s. Try the same thing with 4’s; 
with 5’s; 6’s; 8's; T’s. 
14 
11+ ? =14 !/,of ? =2 210 — 5 a dae ee 
OD As RM erat Yenaets nat pa VA meteor ean te es 
Wa 11 10 4 A eee 
]4-- 2? =7 2Aet = 149) LIS a 14d ie Tsien 
Party O14 14— ? = 100 a 0 ee 
PR Set ey (ait ee © LO A ee 


15 
The S15 215-6 =? 4 SP te 
Meorlos=? «4-2 =156 3x? =15' 1946 =f 
I-Fl4—? Jot? i=3 ?-I18=2 “o9X32 
Li 11. /5 of lO 2, A of LO Se oak se 
Mit Galo ?oSele los 8 ep Terie 
1S S224 3 SHA One Sl db tee 
I3+.%2=15 15-? =13. 8+7 +? <Ie-F2=? 


LESSON XXXI. 113 


Ce 


16 

8+8=2% 16-9=? 16+8=? 7+? =16 
16—? =11 ?+11=16 4+? =16 16—-14=? 
4x4=? 16+? =8 }/s0f16=? ? of 16=8 
pr 4=-16 14+2=? 16-?=9 16-—? =12 
16—? =14 ?Pofl6=d!/, 2x8=? 16+4=! 
of 16=? 8x? =16 ?-8=8 114+? =16 
a+: =16 16-I3=? eof? =2° 16-? =d 


17 

13+4=? 17-?=6 154+?=17 10+? =17 
esos Sh eg ll) on ee 
7+ 4=? 17-15=? ?+5=17 YsoflT=? 
er lisl7 lgotli= 2’ lies 2 5? -Pliskt 
Hi-10=2 ore Sli 14+? =17. li-l4=" 
W,ofli=?° ?-4=13 17-8=? 17+2=? 
Tie your counters together by 2’s, and try to 
make 17 with 2’s. Try the same thing 

with 3’s; with 4’s; with 5’s. 


wake 
10+? =18 “18-6 =?" 16+? =18 18-7? =11 
Peete Ph LOSE REL ON B16! ="? 6x ? =18 
18+ ? =6 9xX2=? 18-?=14 14+4=? 
lit? =18 44+7?=18 3x? =18 1/6 of ? =3 
Me Or is— 7 l8= 4'— PP 12-- Pp =18' 18+ 29 
2X 2? =18 oof l8=? '/,0f18=? Ir 9=? 


114 LESSON XXXII. 


19 
12+7=? -2+7=19 19-6=? 1542 =19 
?—4=15 19-14=? ‘fofl9=? 19-12=? 
19+6=? Yofl9=? 5+7=19 19+ 9=? 
3+9=19 ?-7=12 ?-8=11 13+? =19 
19-2 =18 19+3=? 19+2=? Yeofl9=? 
W,0of19=? ?+8=19 19-?=2 19-?=10 


20 
Ti 720+), DF 2 =O0WE Oe open 
20-552? 20-1822 ?4+4=90 Bx10—r 
4177 TOX=20 BP0-I2=? 20ers 
290-2 =14 18+7=? B+4=? Yrof? =5 
Weot20=) JOP =o oe ofa a Tb eiaeeO 
4x12? =20 20-9 Sl 1545=2 205 =T0 
?-12=8 I, o0f20=? (?-3=17 20-18=? 


APPLICATIONS IN UNEQUAL DIVISIONS. 


1. Carl has a square garden. It is 17 feet around 
the 4 sides of it. How long is one side? 
2. I have 13 cakes to divide among 6 children. 
How much cake willeach one get? 13+6=? 
3. Will, Carl, and Fred have 19 yards of kite- 
string. How much of it belongs to each? 
4. In 7 hours Ned walked 15 miles. How far 
did he walk in 1 hour? 1/; of 15 =? 


LESSON XXXIII. 118 


FRACTIONS. 
Words to be /earned by sound, 


square eXx-préss’ it-sélf” éi'ther 


:. 


2. 


aa 


Draw a square on your slate, making all the 
sides exactly one inch long. 

If you make the sides longer than the ends, 
will it be a square? What will it be? 


. Draw a line across the middle of 


your square, thus, and tell by 
figures and also by words how 
much of the square is above 
the line, and how much of the 
square is below the line ? 
Is either half a square by itself? What is it? 


. Draw another square inch. Draw a line 


across the middle, then draw one down the 
middle, as in this square. 
Into how many parts is this 
square divided?  Hxpress it 
by figures, thus: 4/4. What 
is the shape of these parts ? 
Tell by figures what part of 
the whole square each little square is. 


. In one half of the square how many fourths 


are there? Then 1/9 = 2/4, and 1/5 of 1/4 = 1/4. 


116 LESSON XXXIII (continued). 


8. Erase one small square. How many fourths 
were there in all? How many did you 
erase? How many are left? Express all this 
by figures, thus: 4/4—1/4= 8/4. 

9. How many fourths have been 
erased here? How many 
fourths are left? Express 
by figures;, 4/4 — 2/4=? 

10. How many fourths would you 
have to add to these to have a whole 
square? 2/4? = 4/4, 

Ll. Draw another square and divide it into 


fourths. Erase 3 pate gi cg =? 
(a) 


FADD 


12. Are the parts of these squares of the same 
shape? Of the same size? If they were 
plates of gold, would 1/, of the first cost as 
much as !/, of the second or of the third? 

13. Copy these three squares, and make all the ~ 
problems you can by erasing parts, telling 
by figures and signs all that you do, 


LESSON XXXIV. » opp et 


THIRDS, SIXTHS, AND NINTHS. 


. Draw a square inch on your slate, divide it 
into three equal parts, and express the 
number and the size of the 
parts by figures, thus: °/s. 

. Erase one third, and express by 
figures and signs how many 
there were at first, how 
many erased, and how many 
left. Then 3/s oe 1/s =? 

. How much of this square inch 
has been taken away? How 
much is left? °/;—#/; =? | 
. How many more thirds would ——--—~ 
make a whole inch? Then !/3 +? = 3/s. 

. Draw another square and divide it into thirds. 
Then draw a line across the middle of each 
third. Into how many parts 

does this divide each third ? 

. Into how many parts does it 

divide the whole square? 

Express it in figures, thus: °/¢. 

. In one third, how many sixths 

are there? Then !/3 = 2/g, and 1/2 of 1/3 = 1/6. 
. How many times could you erase ?/¢ at a time? 
3/, at a time? 


13. 


14, 


1d. 


16. 


ive 


LESSON XXXIV _ (continued). 


. Hrase 1/¢, and tell what is left. /, —1/6=? 
. Hrase '/g at a time and tell: %.—2/,=? 


Ver jemi S/iemy/éanpe ‘/oms/ane 


. How many 6ths are ?/g and 1/6? 2/g+4/6=? 


eit lis = 2)\/saridle =a ee ee a 


. Draw another square inch and 


divide it into thirds. Divide 
each third into three equal 
parts, as in this square. Hx- 
press in figures the number 
and size of the parts, thus: °/». 

How many 9ths in each third? Then !/3=3/, 
and 1/s of a oY ald 

How many times could you erase 3/p ? 

How many 9ths are °/y) and 3/9? 5/9 + 4/9 =? 
Plo siit obi Afar P/o ate om> tho laay test) Shiny a ae 
fig t */g=? 

Hrase °/y and tell how many 9ths are 9/y less 
S/o °/o Rot 2eye mal ii dd 99 ome 
Voy "/ 38 elias S/o ena Woe 

Draw three squares; divide one into thirds, 
one into sixths, and the other into ninths, 
and tell: 

How many 6ths are equal to 2/3? Tol/,? 3/6? 

How many 9ths are equal to 2/g? To+4/,? 2/3? 

How many 3ds are equal to 3/9? To 2/6? 


PARI ULV. 


Operations in Numbers above Twenty, 


LESSON TL. 


NOTATION AND NUMERATION IN TENS TO ONE HUNDRED, 


|. Arrange your counters in groups, like these : 
How many tens are there here? 
bh | How many ones ? 
Units is another word to use instead of ones ; it means the same thing. 
Two tens and five units. 
2. Add another ten to your counters. Write how 
many tens and how many units. 
3. Add another ten. Write, as before, how many 
tens you have now, and how many units. 
4. Beside your sentences write the figures which 
express the same thing, thus: t 
Two tens, five ie 2 
Three tens, five units. Be 
Four tens, five units. | 
D. Add five more tens, one ten atatime. Write: 


Five tens, five units. DID 
Six tens, five units. 65 
Seven tens, five units. 75 
Eight tens, five units. 85 


Nine tens, five units. 95 


120 LESSON I (continued). 


6. Take away the five single counters. What 
have you left? How many tens? How 
many units ? 

Nine tens and no units, 90. 
7. Copy the sentences in 4 and J, putting the 


word vo in place of the word five. ra 8 
8. Beside the sentences write the figures 20 
which express the same thing, thus: 30 


9. In these numbers, tell which figures  *e. 
stand for tens, and how many tens. 
20,02, 639° 2 ViSp 405 O36. 82D eos 

10. Read these numbers ; tell first how many tens 
and units, and then give the name: 

20, 17,860,442.) Do; 404.39  26e Oieeous 
135 "6D, 98.719.) Of. 19, AS. SDR Ze ose 

11. Read quickly: 12, 22, 33, 44, 55, 66, 77, 88, 
99, 23, 34, 45, 56, 67, 78, 89, 91. 

12. Read the numbers in 70 and 77, glancing at 
two numbers at a time, and looking off the 
book when saying them. 

13. Write in columns the names and the figures 
for all numbers from fifteen to twenty-five ; 
from twenty-six to thirty-seven; from forty- 
nine to thirty-eight ; from fifty to seventy- 
five; from ninety-nine to seventy-six. 


LESSON II. 121 


ADDITION, IN SUMS BELOW ONE HUNDRED. 


. Place a group of 5 counters on your desk. 

To the right of these place a group of 4 

counters. How many counters 
| | in all are there? Express this 

in figures thus: 5+4=9 

. Place a bundle of counters beside the five. 


What number do you make 

by putting ten to five? ‘pasa Bill | | 
many counters in all? 

. Add another ten to ans pnitee eroup, 

and tell what number you have made. 

25+4=? 

. Add another ten. 35-+4=? Add another ten. 

45+4=? Add anotherten. 5074=? 

. Add another ten. 65+4=? Add another ten. 

75+4=? Add another ten. 85+4=? 

Add another ten. 95+4=? 

. Place your counters in groups of 2 and 6. 

Make all the examples you can of 2+ 6, 

by adding 9 tens, 1 ten at a 

time, to the 2 ones. As you 276=8 

make these additions with 12+6=18 

your counters, write them, 2216=28 


one under another, in the etc., to 
form of a table, thus: 92+6=98 


6 


122 LESSON II (continued). 


7. Place your counters again in groups of 2 and 
6. Add 9 tens, 1 ten at a time, , 
to the 6 ones (instead of to the 2+ 6 =8 
2 ones). Write out these addi- 2+16=18 
tions as you make them, in 2+26= 28 
the form of a table, thus, and etc., to 
compare with the table for 2+ 96 
Ci cab, TAS 6 VAG Tete, 

8. In the same way, make and write two tables 
of additions for 4+3; also, two tables for 
» P25°f0r6 FS for 3b tor teen 


6 + 4. 
9. Arrange your counters in groups of 5 and 7. 
Bre? 


10. Now, make a table of additions by adding 8 
tens, 1 ten at a time, to the 5. Write the 


table as you make it. 

Do not ‘“‘make up the ten” formed by adding 5 and 7, for you 
would have to untie the 10 and arrange again in groups of 7 and 5 
for each new addition. 


11. Inthe same way, make and write a table of 
additions for 8+ 7, and one for 7+ 8, and 
compare the two tables. 

12. In the same way, make and write tables for 
3.7 6eand BS O43 Bfor. oe erie ee 
D4'6 and'6 +5.- fdry9 BS Rlands (ao eon 
8+6 and 6+8. 


——— ee 


12. 
15. 


LESSON III. 123 


CONCRETE APPLICATIONS. 


Carl has 67 cents in the bank and 8 cents in 
his purse. How much in all has he? 

I found 46 carnations this morning, and 9 
this afternoon. How many in all? 

I spent all the month of August and 7 days 
of September in the country. How many 
days in all? 


. Carl paid 46 cents for a fishing-pole and 8 


cents for fish-hooks. How much for both ? 


. Mary is 17 years old and Anna is seven years 


older than Mary. How old is Anna? 


. How many inches long is a board that is a 


foot and 9 inches long ? 


. There are 60 minutes in an hour. How many 


minutes in an hour and a half? 


. May had a quarter of a dollar, and Will gave 


her 9 cents more. How much has she now ? 


. How many hours in one day? How many 


in a day and a half? 


. Will is 2 feet and 9 inches taller than little 


Fred. How many inches is that. 


. Fred is 2 feet and 3 inches tall. How tall, 


then, is Will? 
How many cents in 3 dimes? In 5 dimes? 
How many dimes in $3? In $5? 


124 LESSON? IV: 


OBJECT AND SLATE WORK. 


“Work out with counters, copy and complete. 


7+ 36 = 9 18 yo LS Oo = 7+14= 
si+t+6= 294+ 8= o9+d9= 4+ 7= 
67+ 6 = S219 = 51120 4D ae 4+47= 
1906 7 Gee gosh 9+39= 444+ 7= 


Copy, complete, and prove with counters. 


be 2 Oi A 0 a ted aid 
Li Oo 4 2 at Pe ae 
LOSE Oy (te OO a ears Geta 


3+37= 18+ 3 =) 76+ 5 =. 64 19= 
47+3= 83+8= 5+66= 9649 = 
58+7= 8488= 554+6= 69+6= 


RAPID ORAL CALCULATIONS. 

Read, stating sums only, thus: “8, /6, 32,” ete. 
‘fap tela rtelinmieiaptebom (ey mevelimtel ap eae to: => 
Sas unde bara tle shane ap ore S= 
lime iarellontGar barthieibariin tear is = 


1 fer ttt = su) 8 Yor ne oe Peo Ie 9+18= 
14 2 4+19= Gt Lis “Utd oe 
Me OSD eo Se eS oe ene O+17= 


Begin with 1 and add 4’s to 41. Begin with 3 
and add 4’s to 43. Begin with 2 and add 
5’s to 52. Begin with 2 and add 7’s to 72. 


LESSON V. 125 


SUBTRACTION, IN NUMBERS BELOW ONE HUNDRED, 


. Place 9 counters on your desk; take away 4 
of them. What have you left? 
9 less 4 are 5, 
. Work out these examples with your counters, 
copy them in the form of a table, and com- 
plete: 19-4; 29-4; 39—4; 49—4; 59 
oe ee Oeming: 1. Oh 489) 2 A. GO aA 
. In the same way, work out and write a table 
of subtractions of 8 — 6, etc., to 98 — 6. 
. In the same way, work out and write a table 
of subtractions of 7 — 3, etc., to 97 — 3. 
. In the same way, work out and write a table 
Or subtractions’ of 6°-'D, etc!, to 96— 5: 
also, one of 5—3, etc., to 95 —3; also, of 
4—2 etc. to 94— 2:"of3'— 3, ete, to 93 —s: 
. How many more are 9 counters than 4 count- 
ers? Place 9 counters on your desk ; under 
these place 4 counters. 
. Now, to find the difference between these two 
numbers, we will take as many {|| || 
counters from the greater num- 
ber (9) as there are counters in 
the smaller number (4), and what 
is left will be the difference between them. 
Then 9 are 5 more than 4. 9—4=85, 


126 LESSON V_ (continued). 


8. Notice that we express these two questions, 
“9 less 4 are how many?” and ‘9 is how 
many more than 4?” by figures and signs, 
in exactly the same form, thus: “9—4=?” 

9. In the same way find how many more are 


9,than 6) jev thangdi? GithannZan) saganel 
19 than!6. -27-than.5-- t6 than -2Pisithanal. 
39 than 6 d7than5 S56than2 23 than 1 
69 than6 77 than5 86than2 73 than 1 


CONCRETE APPLICATIONS. 


1. How much older is Anna, who is 19 years old, 
than Frank, who is only 6 years old? 

2. John had 48 marbles, but he has lost half a 
dozen. How many has he left? 

3. There were 97 trees in the park. - The wind 
blew down 6 of them. How many are left ? 

4. I have 32 buttons on a string, but Anna has 
38. How many less have I than Anna? 

9. Our hall is 40 inches wide. Will yard-wide 
carpet cover it from side to side? How 
much wider is the hall than the carpet ? 

6. Nellie found 4 dozen eggs in the barn last 
week, and only 45 this week. How many 
less this week than last ? 


LESSON VI. 


127 


OBJECT AND SLATE WORK, 


Work out with counters, copy, and complete these examples, writing 
the sign — in place of the word “ less,” 


How many are 


lv less4 16less3 i18less5 19 less 7 
27 less4 26less3 28less5 £99 le&S 7 
D7 less4 S8361less3 ‘58 less 5 . 29 less 7 
Wles4 14less3 15les4 #4217 less 6 
Soles 4 7 24 "ess 3°") 25 lesa 4") VT Tess 6 

Sless4 34]less3 35 ]less4 77 less 6 


Copy and complete, writing the sign — in place of the words 
“than” and “and,” 


How many more are 
13than3 18than6 59than5 26 than 4 
33 than3 28 than6 69 than5 46 than 4 
73 than3 98than6 79thand5 86 than 4 


What is the difference between 


9and8 45 and2 38and7 £469 and3 

39 and8 85and2 18and7 79 and3 

5Dand8 Q95and2 58and7 # £89 and 3 
RAPID ORAL CALCULATIONS. 

oT hss 89-3 = 1 26 -— 5 = 9 -— 8 = 

A514) 6 —4— 0 63-328. -98=7= 

a2 —-D =i (Sr ie 9b-1= 67-6= 


128 LESSON VII. 
1. 23 counters less 7 counters are how many ? 
Arranging the counters thus, ff] 
we see at once that we can not [ff a 
take 7 ones from 3 ones, so we untie one 
tl of the tens and arrange 
| | | the counters thus, having 
1 ten and 13 single counters. Now, taking 
the 7 ones from the 13 ones, we find that 
we have six ones and one ten (16) left. 
The work we have done with our counters 
is expressed by figures and signs thus: 
23 —7 = 16. 
2. In the same way, find and express how many 
Ale. 49 i 50 Ol ae One) Ome a cee ie 
IN He Reoee eS LHR 6 teeta Mp ser. 
3. Make with counters and write a table of sub- 
tractions, by taking 7 counters from each 
of these numbers: 15, 25, 35, 45, 55, 68, 
FAD eeCo) eo 
4. In the same way, work out and write tables 
of subtractions for 17 — 9, etc, to°9T — oe 
13-5), 6teeo lai Oy ete epee wee Ghee 
Tl 3, ete. 18h 9) GlCreLo. vee cen 
12 = 4, ete. ; 19 Ll eter eects 
136, etc.; “14 '3) etc, slo ad eo reme 
1673, etc.; “15 9) "etc. ieee re 


LESSON VII (continued). 129 


). How many more are 23 than 7? Place 23 


6. In 


Peake 


counters on your desk; place 7 counters 
under them. To find the difference between 
these two numbers, we must take as many 
counters from the larger number (23) as 
there are counters in the smaller (7). Can 
you take 7 ones from 3 ones? What, then, 
must you first do? Work out the answer 
to this question with counters, and express 
it in figures and signs, thus: 25 — 7 = 16. 
the same way, find and express the differ- 
ence between 24 and 9; 27 and 8; 51 and 6; 
43 and 9; 52 and 5; 66 and 9; 81 and 8. 


CONCRETE APPLICATIONS, 


seven years John will be 34 years old. 
How old is he now? 


2. Carl’s kite-tail was 27 feet long, but 8 feet of 


it were torn off. How much is left ? 


. Nellie had 42 buttons on her button-string ; 


now she has 34. How many has she lost? 


A table is 3 feet wide, and is 7 inches longer 


than it is wide. How many inches long is it? 


. Walter is 8 and his brother is 7 years old. 


What is the difference in their ages? 


130 LESSON VII (continued). 


10. 


. Carl had 55 cents; he spent 9 cents for slate- 


pencils and candy. How much had he left? 


. This afternoon he has spent 8 cents more, for 


writing-paper. How much remains now? 


. Harry and Ned have, together, 43 marbles; 


9 of them are Ned’s. How many are Harry’s? 


. Tom had 385 pop-corn balls to sell on the 


trains. On the 4 o'clock train he sold 8. 
How many had he left? 
On the 4.15 train he sold 9 more. How many 


had he then left? On the 4.30 train Tom - | 


sold all the rest but 9. How many did he 
sell ? 


. Will’s top-cord was | yard, 1 foot long. He 


broke off9 inches. How long was it then? 


. Tom had a quarter of a dollar, and spent 6 


cents. How much was left ? 


. Nell had a basket of 2 dozen eggs; she let 


the basket tip over, and 8 eggs fell out and 
were broken. How many were left? 


. If you sleep 7 hours, how many hours of the 


day are you awake? 


. 9 of the 52 weeks of this year are past; how 


many are still to come? 


. In my garden to-day there were 33 carna- 


tions; I picked 7. How many did I leave? 


butt (Dn) al al 


LESSON VIII. 131 


OBJECT AND SLATE WORK, 


Work out with counters, copy, and complete. 


S07 te 6 Rb 8H lao 

Sur MeO Minh Ose ~~ Sr675 = 

0S  OsThine + om 2S 6S = 
D Speas or 4570-657. 85977 b= 


Copy, complete, and prove with counters. 


30 —-8= 24 —-8= 33 — T= 45 -6= 

72-9= 83 —9= 23 —4= 62—6= 

gl —9= 92 —-3= 91 —-—2= Day Dae 
Do Oi ar 60 10s 6570. nn6 ib = 
iia Ore © Ol 76s 7Ocpbnnb bia = 
Way; O26 aA 2517 8K ee 1-9= 


RAPID ORAL WORK, 
Read, stating remainders only, thus: “80, 27, 24," ete, 


YB PSE og hates Tae Be Soe im ts One 8 ends Manet We 

Bees ee eg a(S} 0) LC) 

SLES a RAE OM Rea Some RU Rae 

Ge P BoPe STP VAP Bie eras reales Meare 0 Woe 
Begin with 40, and take as many 4’s as you can. 
Begin with 38, and take 3’s; 5’s; 7’s; st 
Begin with 54, and take 8’s; 9’s; 2’s; 7’s; 6's. 
Begin with 67, and take 6’s; 8’s; 4’s; Os. rats 2 


132 *LESSON IX. 


MULTIPLICATION. 
Making and Learning the Tables. 


ssi 


. Group your counters by threes until you have 
10 threes. How many counters in all is 
this? Copy and complete each of these 
examples as you find the answer, and 
write in the form of a table. How many 
are three 3’s? Five 3’s? Seven 3’s? Nine 
3's? Ten 3’s? Hight 3’s? Six 3’s? Four 
3s? Two 3's? 

. Arrange your counters in 10 groups of 4 each. 
Find answers, copy, complete, and write as 
a table, how many are two 4's? Four 4’s? 
Six 4’s? Hight 4’s? Ten 4’s? Nine 4’s? 
Seven 4's? Five 4’s? Three 4’s? 

. Arrange your counters in 10 groups of 5 each. 
Find answers, copy, and complete. How 
many are "3 *D2 oO KD? 7 Xora 
DEX DP MASK 9 1) MG) Pie eee Oe 

4, Arrange your counters in 10 groups of 6 each. 
How many are one 6? Two 6’s? Three 6’s? 
Four 6's? Five 6’s? Six 6’s? Seven 6's? 
Hight 6’s? Nine 6’s? Ten 6’s? 

. Arrange your counters in 10 groups of 7 each. 
How many are7X7? 9X7? 3X7? 6X7? 
DIANE 2 PAIX T 2) BOX TOE D OT I TOUR ee 


bo 


ey 


Or 


LESSON IX (continued). 133 


6. Arrange your counters in 10 groups of 8 each. 
How many are two 8's? Four 8's? Hight 
8’s? Three 8’s? Six 8’s? Nine 8's? Five 
8’s? Ten 8’s? Seven 8's? 

7. Arrange your counters in 10 groups of 9 each. 
How many are 10X9? 9x9? 8x9? 
1X9? 6X92 OK 924%97 3X92.2XK 9? 

8. Arrange your counters in 10 groups of 10 each. 
How many are2X10? 3x10? 4X10? 
9X10? 6X10? 7x10? 8x10? 9x10? 
10 x 10? 


DICTATION EXERCISE, 


e@e$88 86 

1. How many dots in this square of °¢e¢¢e 
5 dots each way? 5X5=? HESS Beek x 

2. Make a square of dots, four dots **¢ee¢e 
9 


each way. How many dots inall? 4x4= 


3. Make a square of 7 dots each way. 7 X 7 =? 
4. Make a square of 8 dots each way. 8X 8=? 
5. Make a square of 6 dots each way. 6x 6=? 
6. Make a square of 9 dots each way. 9X 9=? 
7. Make a square of 3 dots each way. 3X3=? 
8. Make a square of 2 dots each way. 2x 2=? 
9. Make asquare of 10 dots each way. 10 x10=? 
10. Complete, and learn this table of squares. 


134 


ee yn pee ed 


LESSON vx: 


CONCRETE APPLICATIONS. 


. John earns $3 a week. How much does he 


earn in 2 months? ($ stands for dollars.) 
How many shoes will it take to shoe 7 horses? 
How many days are there in 8 weeks ? 
How many school-days in 9 weeks ? 


. Ned has 6 quarts of strawberries. How 


many pint-baskets can he fill with them? 
(There are 2 pints in 1 quart.) 


. How many quart-baskets could you fill from 


1 peck of plums? (There are 8 quarts ina 
peck.) How many from 2 pecks? 3 pecks? 


. How many sides have six i Picture this 


and the next 7 examples. 


. How many sides have 9 triangles ?. 

. How many faces have 7 cubes? 

. How many sides have 8 pentagons ? 

. Four pentagons have as many sides as how 


many squares ? 


. Three pentagons have as many sides as how 


many triangles ? 
Three squares have as many sides as how 
many triangles ? 


. Make 12 pentagons, 20 triangles, and 15 


squares, and count the sides in each set 
of figures. 


a". ap 


1X2= 6xX%2= 1X3= 6X38= 1x4= 
Pee A OR EXIS — * Bde 
SBA2Z= 8X2= 38X38=°8x3= 38x4= 
4X2= 9xXx2= 4x3= 9x38= 4x4= 


LESSON XI. 


OBJECT AND SLATE WORK. 
Copy, complete, and /earn, 


135 


5 X4'= 
yee. 
8x4= 
9x4= 


OX2=10K2= 5X3=10X3= 5x4=10x4= 


Work out with counters, copy, complete, and /earn., 


1X5= 6X5= 1X6= 6X6= 1x7T= 
2X5= (X5= 2X6= TX6= 2XT= 
peo SOND os6=> 8X6= 3XT= 
aoe ey AK oO 9X OS AKT SH 


6xX7T= 
TX7T= 
Sxi= 
IX7T= 


en tee tae Os LOG oe = AO Xe 


1X8= 6x8= 


1x9= 6X9= 1X10= 6X10= 


ZxO= (X8= 2x9= TX9= 2X10= TX10= 
BRK ioe io eeo BUS 69 SLO BxahoS 


4X8= 9xX8= 4x9= 


9xXx9= 4x10= 9X10= 


5X8—10X8=.5x*9=10:X%9=.-59%10=10X10= 


six 6’s 
nine 9’s 
eight 8’s 


RAPID ORAL CALCULATIONS. 


9X7T= 7X8= 
8xX4= 6X9= 
>X8= OX T= 


eight 6’s 
nine 8's 
six 7’s 


| 


| 


136 *LESSON XII. 


DIVISION—CASE /, 


1. Arrange 24 counters on your desk; separate 
them into groups of six. How many 6's 
are there? 24+6=? 

2. In the same way, work out with your count- 
ers, copy, and complete these examples: 

Did — "00 4" 20rd OU Danae ee 
96+7= 16+4= 25-5= 3676= 49-7= 
64°38 — Sl=9=' 48-0 =) 04-9 — 606 

3. Also work out with your counters, copy, and 
complete these : 

24 A= BD> Da gde Say 0 = 1. pieces 
28-7= 4075= 5426= 21+7= 32-8= 
63=-T= 49+9= 30+6= 24+8= 28+4= 


CASE //, 


4, Separate 32 counters into 8 equal groups, and 
find and tell how many counters in one of 
these groups. 1/s of 32 =? 

0. Work out with counters in the same way, and 
copy and complete these examples: 


1/5 Cie 1/6 of 138 = =p of 24 = 1 of 27 = 
V/gof 28 = Myof386= '/,of21= 1/,0f 42= 
Vgof56= Yrof68= 1/,of54= 1/)0f 81= 
V,of16= YWoofT2= Ygof35= ',of 72= 


bo 


On 


LESSON XIII. 137 


DICTATION EXERCI/SES, 


. Make 36 dots on your slate, so that there will 


be the same number of dots each way. 
The square of what number is 36? 

Arrange 64 dots in a square. The square of 
what number is 64? 


. Arrange 25 dots in a square. 25 is the square 


of what number ? 


. Arrange 49 dots ina square. 49 is the square 


of what number ? 


. Arrange 81 dots in a square. 81 is the square 


of what number ? 


. Find, by trying, which of these numbers can 


be arranged in squares and which can not: 
39, 40, 16, 22, 12, 9, 24, 48, 49, 88, 72, 64. 


CONCRETE APPLICATIONS. 


Our house is 27 feet wide. How many yards 
wide is that? (3 feet in 1 yard.) 

How many hours do I sleep if I sleep 1/4 of the 
day? (A day is 24 hours.) 

I have studied !/, of an hour. How many 
minutes is that? (An hour is 60 minutes.) 

Carl has 36 marbles, 9 in each of his pockets. 
How many pockets has he ? 


1388 


D. 


6. 


7. 


eh 
2 


10. 


iN 


12. 


13. 


14. 


15. 


16. 


LESSON XIII (continued). 


Nita has set up 45 blocks in 5 rows. How 
many in each row? 

If you make 72 dots in 8 rows, how many 
dots will there be in each row ? 

How many rows would there be if you make 
them rows of 12 each? 

How many rows if in rows of 24 each? 

Carl ate 8 plums, which were !/; of all he had. 
How many had he at first. 

Nita is 6 years old, which is !/; of her moth- 
er’s age. How old is her mother ? 

How many quart bottles can be filled from 
24 pints of milk? (2 pints in 1 quart.) 

How many gallon jugs would be filled from 
32 quarts? (4 quarts in | gallon.) 

John made 40 cents to-day selling pop-corn 
balls at 2 for 5 cents. How many did he 
sell ? 

Thomas earns !/g as much money a month as 
his father does, whose wages are $48 a 
month. How much does Tom earn? 

Aunt Sarah says that when I have earned 40 
cents she will give me !/, as much. How 
much shall I then have? 

I have !/, as many marbles as Tom, who has 
42. How many have I? 


139 


LESSON XIV. 
OBJECT AND SLATE WORK. 
Write the complete answers. 
How many times 
3 in 4 in 5 in 6 in 7 in 
27 36 35 36 49 
30 24. 30 42 42 
21 32 45 54. 56 
How many 
8’s in 9’s in 10’s in 
24: F2 yA ida 43 AO7E5'30 
o2t 00 a LL Bers. 40 60 
rh 64- H4e— 281 80 3690 
40 48 a6+0 48 10 +100 


Work out with counters, copy and complete. 


39-7 = 


36+ 7= 28=-6= Vsof28= 1/, of 44= 

4] +5 = 38 +4 = 2/3 of 33 = = 8/g Of 832 = 
RAPID ORAL CALCULATIONS. 

36 +9 = 32+ 8= 63-9 = 

54 +6 = 49=+-7T= (2a 


ot Pao 


Perorie  */,orae =m ty oh Gee '/pof 56 
1, of 2B= 1, 0f382= %,o0f64= 5/, of 40= 
Ys of 35= %so0f45= Yyof54= 3/5 of 42= 


140 LESSON XV. 


if 


10. 


FRACTIONS. 
Draw a square.inch. Draw a line down the 
middle. Into how many equal parts does 
this divide it? Express this in figures. 


. Divide each half into two equal parts. How 


many parts in all, now? 1/2, 0f !/,=? 


. Now, divide your square as this 


one is divided, by drawing a 
line through the middle of 
each fourth. Into how many 
parts does this divide it? 
Then 1/5 of 1/4 =? 


. How many 8ths in 1/4? 3/4 = how many 8ths? 


1/, = how many 8ths? 


. How many times ?/s are there in the square ? 


Then 8/g “3 2/s 40 ve of 8/g =? 


. How many times */s are there? 1/, of 8/g =? 
. How many more are ®/g than 3/g? %/g+?= 


e/a: ALT mee 3/9 7/3 =? 2/3 +? = 7). 
Sig Vg =? ORES Acerca hy 


. How many 8ths are 8/, less 2/g? 8/g—®/g=? 


8/,— 7/3 =? 


. How many 8ths are 2 times 2/g? 3X 2/g=? 


2X 3/g =? 
How many times ?/g in 8g? 4/,=%/g=? 
Vp of 8/g =? a of 8/g =? 1/5 of 6/g =? 


.- _ "LESSON AVI. 141 


ADDING BY TENS. 


. Do you think it is any easier to add together 
3 single counters and 4 single counters, 
than it is to add 4 bundles of ten and 3 
bundles of ten? ‘Try it and see. 

. How many tens are 4 tens and 5 tens? How 
many single counters in 4 tens? How 
many single counters in 5 tens? Then 40 
and 50 are how many ? 

. How many are 6 tens and 2 tens? 60+20=? 
LO 20 =?) 40-7 30:12 10H 00S? 

. How many are 3 tens and 4 tens and 2 tens? 
30+ 40+ 20=? 20+10+30=? 

. How many are 30+ 20+8? 50+40+9=? 
380+15790=? 401+15740=? 

. How many are 10+ 20+30+ 20+10+7? 

. Place 32 counters on your desk. Under them 
place 26 counters, so that the 2 tens will 
be under the 3 tens, and the 6 singles will 
be under the 2 singles, thus : 

Now add these two numbers i 4 A || 

by putting the units with the i 

units, and the tens with the H) al ] 
tens. What is the new number pee you 
have made? This is called The sum of the 
two numbers. 


142 LESSON XVI (continued). 


8. Express in figures what you had done with 
your counters, by writing the numbers  ;. », 
to be added one under the other, and 32 
then draw a line under them, and 26 
write the sum of the numbers below 58 
the line, thus: 

9. Work out with your counters, and express in. 
figures, as above, these additions: 42 and 
25; 55 and 14; 62 and 27; 81 and 15; 
75 and 22; 33 and 44; 51 and 36; 23 and 
42 and 14; 21 and 34 and 23 and 15. 

10. Place on your desk 47 counters, and under 
them 38 counters, and under these 14 
counters. When you add the 8 units (single 
sticks) to the 7 and the 4 units, what do 
you find? What will you do with the ;,, 
new ten thus made? (Add it in with 47 
the 1 and 3 and 4 tens.) What you 38 
have done with your counters is here 14 
expressed in figures. 99 

11. Add 1 more counter to your 9 single 
counters. You can now make another ten. 
How many tens in all have you? 

12. There is another name for ten tens; do you 
know it? Write 

Ten tens make one hundred. 


LESSON XVII. 143 


ADDITION, NOTATION, AND NUMERATION, IN HUNDREDS, 


1. How many. are 78 and 47? Let us picture 
this example, with dots for the ales 
units and rings for the tens, thus: |o_o 0}. 

2. We add first the 8 units and the 7 [° ° °°" 
units, and find that we have 15 | © ° 
units, or 1 ten and 5 units. Un- 
der the line, in the units’ place, 
we will put the 5 units (dots), 
and add the | ten (ring) to the tens (rings). 
This 1 ten added to the 4 and ,,. 


t. u. 
the 7 tens makes 12 tens, or O90 4 
1 hundred, and 2 tens. aS 
3. We put the 2 tens (rings) in the 0 0 
hens} plaice, and*express the Leys °° be 
hundred by a large ring, which “at 


we put in the hundreds’ place. 
4. The work, when completed, will look like this, 
and it may be expressed in figuresthus: ,,,, 
Do. In this number we can tell which rings 78 
O stand for hundreds and 47 
O which for tens, first by their 555 


O 
Doh: 


size, and second by their place, for 

tens are always in the second place to the 
left, and hundreds are always in the third 
place to the left. 


144 LESSON XVII (continued). 


6. After this we will use only the second of 
these two ways of showing which are units, 
which are tens, and which are hundreds. 
That is, instead of using dots and different 
sized rings, we will use dots only, and let 
their places show whether the dots stand for 
units, or tens, or hundreds. Bhai 

7. So, instead of writing it thus, with |Ol? ie 
dots and rings, we will express |. 
this number by dots only, thus, | | | 
and it may be expressed in figures thus, 235. 

8. Copy and complete these examples in addi- 
tion, and then express them in figures: 


e@e | eee 


9. Express these numbers in words: 

125, (28,809 (367,.590, 982, 260 7541 
315 619 701 400 600 780 708 611 
10. Express in figures: one hundred, twenty-one; 
six hundred, forty-two; eight hundred, 
eighteen; seven hundred, seventy; five hun- 
dred, five; four hundred; three hundred, 

thirty-three ; nine hundred, nineteen. 


LESSON XVIII. 145 


ADDITION, NOTATION, AND NUMERATION, IN THOUSANDS, 


1. You have learned that ten units make one ten, 
and that ten tens make one hundred. Now, 
what do you think ten hundreds make? 

Ten hundreds make one thousand. 

2. Add 5 hundreds, 7 tens, 6 units, 
and 7 hundreds, 4 tens, 8 units. 
Express these numbers by 
dots, thus. Now, adding the 
6 and the 8 units, you find that 
you have 14 units, that is, 1 
ten and 4 units; you put the units under 
the line in the units’ place, and, adding this 
1 ten to the 4 tens and 7 tens, you obtain 
12 tens, which are equal to 1 hundred, and 
2 tens. Putting the tens in the tens’ place, 
and adding this 1 hundred to the 7 hun- 
dreds and 5 hundreds, you find that you 
have 13 hundreds, that is, 1 thousand and 3 
hundreds. Now, put the 3 hundreds in the 
hundreds’ place and the 1 thousand in the 
fourth place to the left, the thousands’ 
place, and the answer—that is, the 
sum of these two numbers—will be, O76 
1 thousand, 3 hundreds, 2 tens, and 148 
4 units, expressed in figures thus: 1,324 


146 LESSON XVIII (continued). 


3. Copy and complete these examples in addi- 
tion, and then express in figures: 


ee |eees)/ eee 
ee | eeeleeeor 


4. Express these numbers in figures: one thou- 
sand, six hundred, fifty-four; five thousand, 
three hundred, ten; four thousand, twenty- 
seven; eight thousand, eight hundred, 
eighty-six ; two thousand, two. 

Oo. Express these numbers in words: 


L276 4.1; 5,720, \uq,6)8026514 %000n: bis, 001 
10,60 4440 1,001 2,220 1,100 
3,030 5,005 1,010 8020 9,001 


6. Ten thousands make “a ten” of thousands, 
just as ten ones make “a ten.” This num- 
ber, 10,856, is read, ten thousand, eight 
hundred, fifty-six. 

7. Read these numbers: 10,000; 20,000; 30,000; 
15,000; 17,000; 12,856; 14,805; 10,010; 
13,927; 15,005; 16,600. 


LESSON XIX. 147 


CONCRETE APPLICATIONS. 


. There are in my garden 35 roses, 27 carna- 
tions, and 42 sweet-peas. How many in all? 
. May has 125 buttons on her button-string, 
and Nita 108. How many have they both? 
. Twelve dozen of any thing are called a gross. 
12x 12=? How many buttons in a gross 
and a half'a gross ? 

my reader there are, on the first page, 123 
words, the same number on the next page, 
and also on the next. How many on these 
3 pages? There are two ways of doing 
this example. See if you can find out for 
yourself what these ways are. 

. In another book there are on one page 237 
words, on the next, 209, on the next, 223, 
on the next, 207, and on the next,. 252 
words. How many words on these 5 pages? 
Can you do this example by both addition 
and multiplication? Why not ? 

. Last summer Mr. Jones raised 380 bushels of 
wheat, 245 bushels of oats, and 897 bush- 
els of corn. How much grain in all? 

. In 1 mile there are 5,280 feet, and in a !/, of 
a mile 1,320 feet. How many feet in a 
mile and a quarter ? 


a 


melt 


148 *LESSON XX. 


OBJECT AND SLATE WORK. 


Work out with counters, and copy and complete. 


45 and 36 = 32, 26, 18, and 30 = 
27 and 39 = 16, 26, 36, and 6 = 


48 and 56 = 12,42, 8 and 36= 
Write in columns and add. | 

19, 9, 13, and 24 13, 15, 16, 17, 18, and 19 

16, 28, 32,and4 12,19, 27, 18, 20, 12, and 8 


Copy and add. 


14h 19) cule 26 16 Av 6 one Sees 
15 Ris 62 JG 8: VES 7am 195 
9 OW Bil 136s 19% 4200 9 
32-32 12 29 6 18 16 
327 693 427 582 1,069 3,686 
502 58 470 399 1,672 4,864 
1,883 12,861 25,005 120,120 
5,972 8542 19,699 22,903 
3,645 7,207 4,805 37,456 

808 20,101 15,062 18.379 


RAPID ORAL CALCULATIONS. 
30 60 20 40 20 47 29 
40 30 50 Li 30 33 29 
20) 9 18 23 38 15 25 


LESSON XXI. 149 


. Draw a square and divide it into 6ths. Then 
divide each 6th in half by , 
drawing a line down the 

middle of it. Into how many 

parts does this divide the 

square? Express this in fig- a 
ures. Then !/, of 1/6 =? 

. How many 12ths in half the square? Then 
peat /iat Titer yh rete = i ays 9 it’s = "ap: 
. How many more are !2/19 than 6/19? "/ig +? = 
ayAL GH bea Atay Rel at ba mil gaa tcf 2 CP 

. How many are !2/19 less 5/2? ~ Less 8/12? 

. How many are 3/1, and 5/19? = "/ig $2? = %/ta. 
4/19 Bila T/h9. 9/19 ttm 4/19. 

. How many 12ths are 3 times 4/12? 2 X 6/19 =? 
4X 3/19 =? OXF /19 GR 3 X 2/p= ? 

. In /;2 how many times 2/12? 4/19? %/19? 3/19? 
1 /o of Bhe=? 1/s of hig =? 
1/6 of 2he=? A Ad of Lah =? 
1/3 of S/ig=? — 1/o of S/n =? 
. In how many different ways 
can you express this much 
of the square-—°/19 ? 

. Answer the same question about 8/12; about 


12 fis; fins 1 /iehoesies) aes 7/1! /aey 4/19. 


150 * LESSON’ CEA. 


SUBTRACTION AND COMPARISON. 


1. Is it any easier to take 4 single counters from 
9 single counters than it is to take 4 tens 
from 9 tens? Try both, and see. 

2. 9 tens less 4 tens =? Then 90 less 40 =? 

3. 8 tens less 5 tens are? 80—90=? 60—20=? 

4. 57 less 34 are how many ? aaa | 
Place 57 counters on | ! 
your desk; from this group of 57 take out 
34, What is left ? 

5). To express this in figures, you write first the 
number of counters you have (57); 4.x. 
then under this write the number (384) 57 
you wish to take away; then, drawing 34 
a line, you write under it the figures 93 
which tell how many of the 57 would 
remain. This is called Zhe Remainder. 

6. 32 less 15 are how many? Arrang- 
ing your counters thus, we see i i al 
at once that we can not take 5 ones from 

i) HII] the 2 ones, so, untying 

i | one of the tens, we ar- . 

range the counters thus, having 2 tens and 

12 ones. Then, taking from these the 1 

ten and 5 ones (15), we find that we have H 

ten and 7 ones (17) left. 


10. 


ie 
12. 


. In the same way, find and express in 72 


LESSON XXII (continued). 151 


. The work we have done with counters 99 


may be expressed in figures thus: ~ 
5 15 


17 
figures and signs how many are 41 
less 14; 37 less 19; 53 less 25. 


. How many more are 637 than 425? Let us 


use dots to find the difference 
between these two numbers. leet. 
Taking as many units, tens, and 
hundreds from the greater num- |). 
ber as there are units, tens, and 
hundreds in the less, we find that 
we have left 2 units, 1 ten, and 2 hun- 
dreds, or 212, which is the difference 495 
between them. ae 
212 
This may be expressed in figures thus: 
Copy, complete, and then express in figures : 


13. Picture with dots, copy, and complete : 


1234 3582 5,796 8092 5,608 
123 401 4563 6,000 3,006 


152 LESSON XXIII. 


CONCRETE APPLICATIONS. 


1. If there are 675 daisies in a field, and Jane 
picks 325, how many will be left ? 

2. Mr. Hudson raised 857 bushels of corn on his 
farm last summer, and has sold 569 bushels. 
How many bushels has he left ? 

3. He also raised 388 bushels of wheat. How 
much more wheat than corn did he raise ? 

4. In the flower-show there were 1,587 yellow 
flowers; 407 were roses. How many of 
other kinds of flowers ? 

9. There were 500 white flowers. How many 
more yellow than white flowers ? 

6. There were 859 red flowers. How many 
more yellow than red flowers? How many 
more red than white flowers ? 

7. I have walked 3,160 feet. How many more 
feet must I walk to make a mile? How 
many feet in a mile. 

8. Will has walked 4,085 feet. How much 
farther has he gone than I? How many 
more feet must he walk to make a mile? . 

9. Frank’s father made $9,875 last year, and 
spent $6,750. How much did he save ? 

10. He made $1,790 less this year than last. 
What did he make this year ? 


*LESSON XXIV. 


OBJECT AND SLATE WORK, 


153 


Work out with counters, and copy and complete these 


Subtractions. 
47 D4 62 48 36 
33 37 20 29 16 
_Comparisons. 
40 65 79 86 49 
25 40) 37 68 37 


Picture with dots, and copy and complete, 


146 269 30D 627 34 
2 «136 = 1038-81982 


Copy and complete, 


427 862 948 1,327 
116 502 908 427 
12,678 15,654. 28,028 
2.564 5,432 8,020 
89,956 65,902 99,090 
9,056 60,092 9.090 


RAPID ORAL CALCULATIONS, 
Read thus: “twenty from forty, twenty,” ete, 
40 60 08 86 330 
2 40 «50 40120 


2,849 
1,627 


39,267 


7,267 


127,342 


79,563 


DD6 
246 


154 *LESSON XXV. 


MULTIPLICATION—A SHORT METHOD OF ADDITION. 


1. Suppose we are asked to find out how many 
trees there would be in an orchard of 6 
rows of trees with 28 trees in each row? 
One way to find out is by Addition; that is, 
to write down the number of trees in 
each row (28) as many times as there 28 
are rows of trees (6). We would 28 
then add, first, the units, thus: “8,16, 28 
24, 32, 40, 48 units—4 tens and 8 28 
units’; putting down the 8 units, we 28 
would then add the 4 tens to the col- 28 
umn of tens and add thus: “4, 6,8, 168 
10, 12, 14, 16 tens—1 hundred and 6 
tens”’—which we write in their proper 
places, having for our answer 168 trees. 

2. Another and a much shorter way to find out 
the same thing is by Multiplication; that is, 
instead of writing down the 28 six times, to 
write it only one time, and under it to write 
the figure 6, to show how many , 
times 28 trees there are; then 28 trees 
we multiply 28 by 6, thus, saying — 6 
“6 times 8 (or 6 8’s) are 48—4 168 trees 
tens and 8 units”—we write 
the 8 units in their proper place; then, over 


LESSON XXV (continued). 155 


the 28 we place a small figure 4, to remind 
ourselves that we have these 4 tens to add 
to the other tens; then, after multiplying 
the tens, saying “6 times 2 tens are 12 tens,” 
we add in the 4 tens, which gives us 16 
tens, that is, 1 hundred, 6 tens, which we 
write in their proper places, and thus show 
that 6 X 28 trees = 168 trees. 

3. Do this example both by addition and by mul- 
tiplication. There are 8 rows of leaves in 
our hall carpet, and 24 leaves in each row; 
how many leaves in all? 

4. Write 43 seven times and add; multiply 43 
by 7. Add 72 five times; multiply 72 by 5. 

). Do these examples both ways: 

38 39 27 124 672 987 


— + 


-_———---—- —--— 


———_—_____— 


pee 9 tr Shad Fad oiling 
536 881 269 380 1,246 
eri ack nctct 
6. Copy and complete these examples: 

2,579 3,520 4,408 5,045 
mies usd Ue bas ad 
15,232 25,550 32,003 51,020 

6 7 3 9 


— 


156 LESSON XXVI. 


10. 


CONCRETE APPLICATIONS. 


John’s grandfather is 3 score and 10 years 
old. What is hisage? (A score is 20.) 


. John’s father is 2 score and 5 years old, and 


John is 5 years less than a score. What 
are their ages ? 


. In a pound of sugar, or of flour, there are 16 


ounces. How many ounces in 8 pounds? 


. There are 100 pounds in a hundred-weight. 


How many hundred-weight and how many 
pounds over in a barrel of flour (which 
weighs 196 pounds) ? 


. Our horse is 15 hands high. How many 


inches is that (the hand-measure is 4 inches)? 


. There are 2 pints in a quart, 8 quarts ina 


peck, and 4 pecks in a bushel. How many 
pints in 1 bushel? 


. How many pint boxes could be filled from 2 


bushels of strawberries ? 


. There are 3651/, days in a year. How many 


days in 4 years? In 6 years? In 8 years? 


. If there are 4 gills in a pint of vinegar, and 2 


pints in a quart, and 4 quarts in a gallon, 
how many gi//s in a gallon? 

How many gill bottles could be filled from 3 
gallons of essence of lemon? 


LESSON XXVII. 157 


DIVISION. 
1. Divide 6 units by 3; divide 6 tens by 3; divide 
6 hundreds by 3. Thus: 
6+3=2 60 + 3= 20 600 + 3 = 200 
2. Here is another form of expressing this same 


thing. It is a better 3)6, 3)60, 3)600. 
TOCRLOSUSE Ie CLYICINID a 6 Sa yo 
1 2 ROOT S200 
arge numbers. 
3. Copy and complete 3)9, 3)90, 3)900 ; 2)8, 2)80, 
2)800; 4)16, 4)160, 4)1,600. 
4, 888 divided by 2 equals what ? 


888 is equal to 5. Here isa short- 
8 hundreds, 2)800 er way of 
8 tens, 80 showing 
SUITS, St ey-ihtF 8 the same 
800 divided by2= 400 thing : 
80 divided by2= 40 
8 divided by 2 = a eee 


me a +44 
888 divided by 2= 444 


6. Do each of these examples in both ways: 999 
+3; 666+2; 444+2; 88874; 555-5; 
848 +4; 426+ 2; 986+3; 248+ 2. 

7. Do these examples in the shorter way only: 333 
+3; 888+8; 666+3; T77+7; 444-1; 
224+2; 363+3; 624 + 2; 663 + 3; 882 = 2. 


158 


a: 


10. 


11. 


LESSON XXVIII. 


CONCRETE APPLICATIONS, 


Carl has been writing for 80 minutes, half the 
time on his slate and half in his copy-book. 
How many minutes has he been writing in 


his copy-book ? 


. There are 600 minutes in ten hours. How 


many minutes in !/; that time? 


. A ranchman has 888 sheep divided into 8 


flocks. How many sheep in each flock. 


. In a tulip-garden of 7 beds there are 777 


plants, an equal number in each bed. How 
many is that? 


. Inarow of ten houses there are 90 windows. 


How many windows to each house ? 


. In 666 lead-pencils how many packages of a 


half dozen ? 


. In 464 shoes how many pairs ? 
. How many horses can be shod all round with 


848 shoes ? 


. | have made a certain number of triangles, 


and find that there are in all 393 sides. 
How many triangles have I made? 

I have 550 ¢ in 5-cent pieces. How many 5 ¢ 
pleces ? 

There are 488 carriage-wheels in a shop. 
How many carriages will they supply. 


LESSON XXIX. 159 


. Whatis!/, of 90? 9 bundles of ten divided 1 in- 

to 2 equal groups i i) i ii 
gives 4 tens in |] ff if 
each group and 1 ten over. Unig the 
ten left over, we divide it in half and find 
that there are five sticks in each half, 
i Hi | | which, added to the 2)90 
ii. tens, gives 45 sticks ~““— 
in each group. Then 1/2 of 90 = 45. 5 
. Suppose we wanted to find 1/2 of 94. One way 
would be to find, first, 2)90 
the half of 90, just as 4 
we did before, and petit : 
then find the half of " oe ee 
the 4 and add it to the ; : nit es 
half of the 90, thus: /2°° 94 sives 47 
. But a shorter way would be to find the half 
of the 9 tens, and, then adding the sticks of 
the 1 bundle of tens left over to the 204 
4 sticks, find at once the half of all ~—— 
the single sticks, thus: #1 
. Do each of these examples both ways: 70 + 2; 
nee 24, Use oO 2A SO O85 +O. 
. Do these the shorter way only: 45+3; 72 
~63 S13; 95+5; 68-43) 844.6; 56 
+2; 64+4; 92-4. 


160 LESSON XXX. 


1. What is '/3 of 729? We will take 7 bundles 
of one hundred, 2 bundles of ten, and 9 
single sticks. 

2. Dividing the 7 bundles of a hundred into 3 
equal groups or thirds, we find 3)700 
that we have 2 bundles in each 
group and | hundred over. In this 9 
bundle of a hundred there are 10 


tens; adding these to the 2 tens we or 
find that we have 12 tens. Wethen 

pe 3" 
divide these into three equal groups, 943 


and find that there are 4 tens in each 
group, and the 9 single sticks divided into 
thirds gives 3 sticks to each third. Now, 
adding together the 2 hundreds, 4 tens, and 
3 units of each third, we find that !/, of 
729 is 243. The work we have done with 
counters is here expressed in figures. ’ 
3. Here is the shorter way of expressing 3)/29 
this: 943 
4, Work out with counters and express 
in both forms these examples: 746 +3; 
D4 4°) 546 D2 028 = 4a (O60 ae 
906+2; T65+3; 948+4; T6595; 
864 + 6. 


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Spencer’s Education: 


INTELLECTUAL, MORAL, AND PHYSICAL. Divided into four 
chapters: What Knowledge is ot most Worth ?—Intellectual Education 
—Morai Education—Physical Education. It is a plea for Nature in 
education, and a protest against tutorial aggression and meddlesome 
overdoing on the part of teachers and parents. Price, $1.25. 


Bain’s Education as a Science. 


The author views the ‘teaching art”’ from a scientific point of view, 
and tests ordinary experiences by bringing them to the criterion of 
psychological law. Price, $1.75. 


Johonnot’s Principles and Practice of Teaching. 


This is a practical book by an experienced teacher. The subject of 
education is treated in a systematic and comprehensive manner, and 
shows how rational processes may be substituted for school-room rou- 
tine. Price, $1.50. 


Baldwin’s Art of School Management. 


This is a very helpful hand-book for the teacher. He will find it full 
of practical suggestions in regard to all the details of school-room work, 
and how to manage it to best advantage. Price, $1.50. 


Bain’s Moral Science. 


A COMPENDIUM OF ETHICS. Divided into two divisions. The 
first—the Theory of Ethics—treats at length of the two great ques- 
tions, the ethical standard and the moral faculty; the second division 
—on the Ethical Systems—is a full detail of all the systems, ancient 
and modern, by conjoined abstract and summary. Price, $1.50. 


Choate’s Elements of English Speech. 


The simple principles of the science of the English language are here 
clearly explained. It is a book designed not so much as a text-book 
as to encourage the study of our language more critically in its forms 
and elements. Price, $1.00. 


Hodgson’s Errors in the Use of English. 
This is a work for the teacher’s table, and invaluable for classes in 
grammar and literature. There is no teacher who will not derive great 
benefit from the careful study of this book. Price, $1.50. 


Sent, post-paid, to any address on receipt of price. 


Descriptive Catalogue mailed free on application. Special prices will be made 
to Teachers’ Reading Circles. 


D. APPLETON & CO., PuBiisHERs, 
New York, Boston, Chicago, Atlanta, San Francisco. 


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